Chapter 8: Polynomials | Turan Arıcan

VIII

CHAPTER 8 Polynomials

Architects use polynomials to design curved shapes such as this suspension bridge, the Silver Jubilee bridge in Halton, England.

This is a photo of a bridge at sunset.

We have seen that the graphs of linear equations are straight lines. Graphs of other types of equations, called polynomial equations, are curves, like the outline of this suspension bridge. Architects use polynomials to design the shape of a bridge like this and to draw the blueprints for it. Engineers use polynomials to calculate the stress on the bridge’s supports to ensure they are strong enough for the intended load. In this chapter, you will explore operations with and properties of polynomials.

8.1 Add and Subtract Polynomials

Learning Objectives

By the end of this section, you will be able to:

Identify polynomials, monomials, binomials, and trinomials
Determine the degree of polynomials
Add and subtract monomials
Add and subtract polynomials
Evaluate a polynomial for a given value

Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form $ax to the m$ , where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,-2x to the 2,4y to the 3$ , and $11z to the 7$ .

Monomials

A monomial is a term of the form $ax to the m$ , where $a$ is a constant and $m$ is a positive whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Polynomials

polynomial—A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

monomial—A polynomial with exactly one term is called a monomial.
binomial—A polynomial with exactly two terms is called a binomial.
trinomial—A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

$mathematical expression$

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

EXAMPLE 1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

$4y to the 2-8y-6$
$-5a to the 4b to the 2$
$2x to the 5-5x to the 3-9x to the 2+3x+4$
$13-5m to the 3$
$q$

Solution

Polynomial

Number of terms

Type

$4y to the 2-8y-6$

$3$

Trinomial

$-5a to the 4b to the 2$

$1$

Monomial

$2x to the 5-5x to the 3-9x to the 2+3x+4$

$5$

Polynomial

$13-5m to the 3$

$2$

Binomial

$q$

$1$

Monomial

TRY IT 1.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

a) $5b$ b) $8y to the 3-7y to the 2-y-3$ c) $-3x to the 2-5x+9$ d) $81-4a to the 2$ e) $-5x to the 6$

Show answer

a) monomial b) polynomial c) trinomial d) binomial e) monomial

TRY IT 1.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

a) $27z to the 3-8$ b) $12m to the 3-5m to the 2-2m$ c) $5 over 6$ d) $8x to the 4-7x to the 2-6x-5$ e) $-n to the 4$

Show answer

a) binomial b) trinomial c) monomial d) polynomial e) monomial

Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

EXAMPLE 2

Find the degree of the following polynomials.

$10y$
$4x to the 3-7x+5$
$-15$
$-8b to the 2+9b-2$
$8xy to the 2+2y$

Solution

a)
The exponent of $y$ is one. $y=y to the 1$

$10y$
The degree is 1.

b)
The highest degree of all the terms is 3.

$4x to the 3-7x+5$
The degree is 3.

c)
The degree of a constant is 0.

$-15$
The degree is 0.

d)
The highest degree of all the terms is 2.

$-8b to the 2+9b-2$
The degree is 2.

e)
The highest degree of all the terms is 3.

$8xy to the 2+2y$
The degree is 3.

EXAMPLE 2.1

Find the degree of the following polynomials:

a) $-15b$ b) $10z to the 4+4z to the 2-5$ c) $12c to the 5d to the 4+9c to the 3d to the 9-7$ d) $3x to the 2y-4x$ e) $-9$

Show answer

a) $1$ b) $4$ c) $12$ d) 3 e) 0

TRY IT 2.2

Find the degree of the following polynomials:

a) $52$ b) $a to the 4b-17a to the 4$ c) $5x+6y+2z$ d) $3x to the 2-5x+7$ e) $-a to the 3$

Show answer

a) $0$ b) $5$ c) $1$ d) 2 e) 3

Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

EXAMPLE 3

Add: $25y to the 2+15y to the 2$ .

Solution

	$25y to the 2+15y to the 2$
Combine like terms.	$40y to the 2$

TRY IT 3.1

Add: $12q to the 2+9q to the 2$ .

Show answer

$21q to the 2$

TRY 3.2

Add: $-15c to the 2+8c to the 2$ .

Show answer

$-7c to the 2$

EXAMPLE 4

Subtract: $16p-(-7p)$ .

Solution

	$16p-(-7p)$
Combine like terms.	$23p$

TRY IT 4.1

Subtract: $8m-(-5m)$ .

Show answer

$13m$

TRY IT 4.2

Subtract: $-15z to the 3-(-5z to the 3)$ .

Show answer

$-10z to the 3$

Remember that like terms must have the same variables with the same exponents.

EXAMPLE 5

Simplify: $c to the 2+7d to the 2-6c to the 2$ .

Solution

	$c to the 2+7d to the 2-6c to the 2$
Combine like terms.	$-5c to the 2+7d to the 2$

TRY IT 5.1

Add: $8y to the 2+3z to the 2-3y to the 2$ .

Show answer

$5y to the 2+3z to the 2$

TRY IT 5.2

Add: $3m to the 2+n to the 2-7m to the 2$ .

Show answer

$-4m to the 2+n to the 2$

EXAMPLE 6

Simplify: $u to the 2v+5u to the 2-3v to the 2$ .

Solution

	$u to the 2v+5u to the 2-3v to the 2$
There are no like terms to combine.	$u to the 2v+5u to the 2-3v to the 2$

TRY IT 6.1

Simplify: $m to the 2n to the 2-8m to the 2+4n to the 2$ .

Show answer

There are no like terms to combine.

TRY IT 6.2

Simplify: $pq to the 2-6p-5q to the 2$ .

Show answer

There are no like terms to combine.

Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

EXAMPLE 7

Find the sum: $(5y to the 2-3y+15)+(3y to the 2-4y-11)$ .

Solution

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.

TRY IT 7.1

Find the sum: $(7x to the 2-4x+5)+(x to the 2-7x+3)$ .

Show answer

$8x to the 2-11x+1$

TRY IT 7.2

Find the sum: $(14y to the 2+6y-4)+(3y to the 2+8y+5)$ .

Show answer

$17y to the 2+14y+1$

EXAMPLE 8

Find the difference: $(9w to the 2-7w+5)-(2w to the 2-4)$ .

Solution


Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

TRY IT 8.1

Find the difference: $(8x to the 2+3x-19)-(7x to the 2-14)$ .

Show answer

$15x to the 2+3x-5$

TRY IT 8.2

Find the difference: $(9b to the 2-5b-4)-(3b to the 2-5b-7)$ .

Show answer

$6b to the 2+3$

EXAMPLE 9

Subtract: $(c to the 2-4c+7)$ from $(7c to the 2-5c+3)$ .

Solution



Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

TRY IT 9.1

Subtract: $(5z to the 2-6z-2)$ from $(7z to the 2+6z-4)$ .

Show answer

$2z to the 2+12z-2$

TRY IT 9.2

Subtract: $(x to the 2-5x-8)$ from $(6x to the 2+9x-1)$ .

Show answer

$5x to the 2+14x+7$

EXAMPLE 10

Find the sum: $(u to the 2-6uv+5v to the 2)+(3u to the 2+2uv)$ .

Solution

	$(u to the 2-6uv+5v to the 2)+(3u to the 2+2uv)$
Distribute.	$u to the 2-6uv+5v to the 2+3u to the 2+2uv$
Rearrange the terms, to put like terms together.	$u to the 2+3u to the 2-6uv+2uv+5v to the 2$
Combine like terms.	$4u to the 2-4uv+5v to the 2$

EXAMPLE 10.1

Find the sum: $(3x to the 2-4xy+5y to the 2)+(2x to the 2-xy)$ .

Show answer

$5x to the 2-5xy+5y to the 2$

EXAMPLE 10.2

Find the sum: $(2x to the 2-3xy-2y to the 2)+(5x to the 2-3xy)$ .

Show answer

$7x to the 2-6xy-2y to the 2$

EXAMPLE 11.1

Find the difference: $(p to the 2+q to the 2)-(p to the 2+10pq-2q to the 2)$ .

Solution

	$(p to the 2+q to the 2)-(p to the 2+10pq-2q to the 2)$
Distribute.	$p to the 2+q to the 2-p to the 2-10pq+2q to the 2$
Rearrange the terms, to put like terms together.	$p to the 2-p to the 2-10pq+q to the 2+2q to the 2$
Combine like terms.	$-10pq to the 2+3q to the 2$

TRY IT 11.1

Find the difference: $(a to the 2+b to the 2)-(a to the 2+5ab-6b to the 2)$ .

Show answer

$-5ab-5b to the 2$

TRY IT 11.2

Find the difference: $(m to the 2+n to the 2)-(m to the 2-7mn-3n to the 2)$ .

Show answer

$4n to the 2+7mn$

EXAMPLE 12

Simplify: $(a to the 3-a to the 2b)-(ab to the 2+b to the 3)+(a to the 2b+ab to the 2)$ .

Solution

	$(a to the 3-a to the 2b)-(ab to the 2+b to the 3)+(a to the 2b+ab to the 2)$
Distribute.	$a to the 3-a to the 2b-ab to the 2-b to the 3+a to the 2b+ab to the 2$
Rearrange the terms, to put like terms together.	$a to the 3-a to the 2b+a to the 2b-ab to the 2+ab to the 2-b to the 3$
Combine like terms.	$a to the 3-b to the 3$

TRY IT 12.1

Simplify: $(x to the 3-x to the 2y)-(xy to the 2+y to the 3)+(x to the 2y+xy to the 2)$ .

Show answer

$x to the 3-y to the 3$

TRY IT 12.2

Simplify: $(p to the 3-p to the 2q)+(pq to the 2+q to the 3)-(p to the 2q+pq to the 2)$ .

Show answer

$p to the 3-2p to the 2q+q to the 3$

Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

EXAMPLE 13

Evaluate $5x to the 2-8x+4$ when

$x=4$
$x=-2$
$x=0$

Solution

a) $x=4$


Simplify the exponents.
Multiply.
Simplify.

b) $x=-2$


Simplify the exponents.
Multiply.
Simplify.

c) $x=0$


Simplify the exponents.
Multiply.
Simplify.

TRY IT 13.1

Evaluate: $3x to the 2+2x-15$ when

$x=3$
$x=-5$
$x=0$

Show answer

a) $18$ b) $50$ c) $-15$

TRY IT 13.2

Evaluate: $5z to the 2-z-4$ when

$z=-2$
$z=0$
$z=2$

Show answer

a) $18$ b) $-4$ c) $14$

EXAMPLE 14

The polynomial $-16t to the 2+250$ gives the height of a ball $t$ seconds after it is dropped from a 250 foot tall building. Find the height after $t=2$ seconds.

Solution

	$-16t to the 2+250$
Substitute $t=2$ .	$-16(2) to the 2+250$
Simplify.	$-16 times 4+250$
Simplify.	$-64+250$
Simplify.	$186$
	After 2 seconds the height of the ball is 186 feet.

TRY IT 14.1

The polynomial $-16t to the 2+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t=0$ seconds.

Show answer

$250$

TRY IT 14.2

The polynomial $-16t to the 2+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t=3$ seconds.

Show answer

$106$

EXAMPLE 15

The polynomial $6x to the 2+15xy$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x=4$ feet and $y=6$ feet.

Solution



Simplify.
Simplify.
Simplify.
	The cost of producing the box is $456.

TRY IT 15.1

Show answer

$576

TRY IT 15.2

Show answer

$750

Access these online resources for additional instruction and practice with adding and subtracting polynomials.

Key Concepts

Monomials
- A monomial is a term of the form $ax to the m$ , where $a$ is a constant and $m$ is a whole number
Polynomials
- polynomial—A monomial, or two or more monomials combined by addition or subtraction is a polynomial.
- monomial—A polynomial with exactly one term is called a monomial.
- binomial—A polynomial with exactly two terms is called a binomial.
- trinomial—A polynomial with exactly three terms is called a trinomial.
Degree of a Polynomial
- The degree of a term is the sum of the exponents of its variables.
- The degree of a constant is 0.
- The degree of a polynomial is the highest degree of all its terms.

Glossary

binomial: A binomial is a polynomial with exactly two terms.

degree of a constant: The degree of any constant is 0.

degree of a polynomial: The degree of a polynomial is the highest degree of all its terms.

degree of a term: The degree of a term is the exponent of its variable.

monomial: A monomial is a term of the form $ax to the m$ , where $a$ is a constant and $m$ is a whole number; a monomial has exactly one term.

polynomial: A polynomial is a monomial, or two or more monomials combined by addition or subtraction.

standard form: A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.

trinomial: A trinomial is a polynomial with exactly three terms.

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Practice Makes Perfect

Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

a) $81b to the 5-24b to the 3+1$
b) $5c to the 3+11c to the 2-c-8$
c) $14 over 15y+1 over 7$
d) 5
e) $4y+17$

a) $x to the 2-y to the 2$
b) $-13c to the 4$
c) $x to the 2+5x-7$
d) $x to the 2y to the 2-2xy+8$
e) 19

a) $8-3x$
b) $z to the 2-5z-6$
c) $y to the 3-8y to the 2+2y-16$
d) $81b to the 5-24b to the 3+1$
e) $-18$

a) $11y to the 2$
b) $-73$
c) $6x to the 2-3xy+4x-2y+y to the 2$
d) $4y+17$
e) $5c to the 3+11c to the 2-c-8$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

a) $6a to the 2+12a+14$
b) $18xy to the 2z$
c) $5x+2$
d) $y to the 3-8y to the 2+2y-16$
e) $-24$

a) $9y to the 3-10y to the 2+2y-6$
b) $-12p to the 4$
c) $a to the 2+9a+18$
d) $20x to the 2y to the 2-10a to the 2b to the 2+30$
e) 17

a) $14-29x$
b) $z to the 2-5z-6$
c) $y to the 3-8y to the 2+2y-16$
d) $23ab to the 2-14$
e) $-3$

a) $62y to the 2$
b) 15
c) $6x to the 2-3xy+4x-2y+y to the 2$
d) $10-9x$
e) $m to the 4+4m to the 3+6m to the 2+4m+1$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

9. $7x to the 2+5x to the 2$	10. $4y to the 3+6y to the 3$
11. $-12w+18w$	12. $-3m+9m$
13. $4a-9a$	14. $-y-5y$
15. $28x-(-12x)$	16. $13z-(-4z)$
17. $-5b-17b$	18. $-10x-35x$
19. $12a+5b-22a$	20. $14x-3y-13x$
21. $2a to the 2+b to the 2-6a to the 2$	22. $5u to the 2+4v to the 2-6u to the 2$
23. $xy to the 2-5x-5y to the 2$	24. $pq to the 2-4p-3q to the 2$
25. $a to the 2b-4a-5ab to the 2$	26. $x to the 2y-3x+7xy to the 2$
27. $12a+8b$	28. $19y+5z$
29. Add: $4a,-3b,-8a$	30. Add: $4x,3y,-3x$
31. Subtract $5x to the 6from-12x to the 6$ .	32. Subtract $2p to the 4from-7p to the 4$ .

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

33. $(5y to the 2+12y+4)+(6y to the 2-8y+7)$	34. $(4y to the 2+10y+3)+(8y to the 2-6y+5)$
35. $(x to the 2+6x+8)+(-4x to the 2+11x-9)$	36. $(y to the 2+9y+4)+(-2y to the 2-5y-1)$
37. $(8x to the 2-5x+2)+(3x to the 2+3)$	38. $(7x to the 2-9x+2)+(6x to the 2-4)$
39. $(5a to the 2+8)+(a to the 2-4a-9)$	40. $(p to the 2-6p-18)+(2p to the 2+11)$
41. $(4m to the 2-6m-3)-(2m to the 2+m-7)$	42. $(3b to the 2-4b+1)-(5b to the 2-b-2)$
43. $(a to the 2+8a+5)-(a to the 2-3a+2)$	44. $(b to the 2-7b+5)-(b to the 2-2b+9)$
45. $(12s to the 2-15s)-(s-9)$	46. $(10r to the 2-20r)-(r-8)$
47. Subtract $(9x to the 2+2)$ from $(12x to the 2-x+6)$ .	48. Subtract $(5y to the 2-y+12)$ from $(10y to the 2-8y-20)$ .
49. Subtract $(7w to the 2-4w+2)$ from $(8w to the 2-w+6)$ .	50. Subtract $(5x to the 2-x+12)$ from $(9x to the 2-6x-20)$ .
51. Find the sum of $(2p to the 3-8)$ and $(p to the 2+9p+18)$ .	52. Find the sum of $(q to the 2+4q+13)$ and $(7q to the 3-3)$ .
53. Find the sum of $(8a to the 3-8a)$ and $(a to the 2+6a+12)$ .	54. Find the sum of $(b to the 2+5b+13)$ and $(4b to the 3-6)$ .
55. Find the difference of $(w to the 2+w-42)$ and $(w to the 2-10w+24)$ .	56. Find the difference of $(z to the 2-3z-18)$ and $(z to the 2+5z-20)$ .
57. Find the difference of $(c to the 2+4c-33)$ and $(c to the 2-8c+12)$ .	58. Find the difference of $(t to the 2-5t-15)$ and $(t to the 2+4t-17)$ .
59. $(7x to the 2-2xy+6y to the 2)+(3x to the 2-5xy)$	60. $(-5x to the 2-4xy-3y to the 2)+(2x to the 2-7xy)$
61. $(7m to the 2+mn-8n to the 2)+(3m to the 2+2mn)$	62. $(2r to the 2-3rs-2s to the 2)+(5r to the 2-3rs)$
63. $(a to the 2-b to the 2)-(a to the 2+3ab-4b to the 2)$	64. $(m to the 2+2n to the 2)-(m to the 2-8mn-n to the 2)$
65. $(u to the 2-v to the 2)-(u to the 2-4uv-3v to the 2)$	66. $(j to the 2-k to the 2)-(j to the 2-8jk-5k to the 2)$
67. $(p to the 3-3p to the 2q)+(2pq to the 2+4q to the 3)$ $-(3p to the 2q+pq to the 2)$	68. $(a to the 3-2a to the 2b)+(ab to the 2+b to the 3)$ $-(3a to the 2b+4ab to the 2)$
69. $(x to the 3-x to the 2y)-(4xy to the 2-y to the 3)$ $+(3x to the 2y-xy to the 2)$	70. $(x to the 3-2x to the 2y)-(xy to the 2-3y to the 3)$ $-(x to the 2y-4xy to the 2)$

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

71. Evaluate $8y to the 2-3y+2$ when: a) $y=5$ b) $y=-2$ c) $y=0$	72. Evaluate $5y to the 2-y-7$ when: a) $y=-4$ b) $y=1$ c) $y=0$
73. Evaluate $4-36x$ when: a) $x=3$ b) $x=0$ c) $x=-1$	74. Evaluate $16-36x to the 2$ when: a) $x=-1$ b) $x=0$ c) $x=2$
75. A painter drops a brush from a platform 75 feet high. The polynomial $-16t to the 2+75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t=2$ seconds.	76. A girl drops a ball off a cliff into the ocean. The polynomial $-16t to the 2+250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall cliff. Find the height after $t=2$ seconds.
77. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $-4p to the 2+420p$ . Find the revenue received when $p=60$ dollars.	78. A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial $-4p to the 2+420p$ . Find the revenue received when $p=90$ dollars.

Everyday Math

79. Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of $x$ miles per hour is given by the polynomial $-1 over 150x to the 2+1 over 3x$ . Find the fuel efficiency when $x=30 mph$ .	80. Stopping Distance The number of feet it takes for a car traveling at $x$ miles per hour to stop on dry, level concrete is given by the polynomial $0.06x to the 2+1.1x$ . Find the stopping distance when $x=40mph$ .
81. Rental Cost The cost to rent a rug cleaner for $d$ days is given by the polynomial $5.50d+25$ . Find the cost to rent the cleaner for 6 days.	82. Height of Projectile The height (in feet) of an object projected upward is given by the polynomial $-16t to the 2+60t+90$ where $t$ represents time in seconds. Find the height after $t=2.5$ seconds.
83. Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial $9 over 5c+32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c=65$ °.

Writing Exercises

84. Using your own words, explain the difference between a monomial, a binomial, and a trinomial.	85. Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.
86. Ariana thinks the sum $6y to the 2+5y to the 4$ is $11y to the 6$ . What is wrong with her reasoning?	87. Jonathan thinks that $1 over 3$ and $1 over x$ are both monomials. What is wrong with his reasoning?

Answers

1. a) trinomial b) polynomial c) binomial d) monomial e) binomial	3. a) binomial b) trinomial c) polynomial d) trinomial e) monomial
5. a) 2 b) 4 c) 1 d) 3 e) 0	7. a) 1 b) 2 c) 3 d) 3 e) 0
9. $12x to the 2$	11. $6w$
13. $-5a$	15. $40x$
17. $-22b$	19. $-10a+5b$
21. $-4a to the 2+b to the 2$	21. $-4a to the 2+b to the 2$
25. $a to the 2b-4a-5ab to the 2$	27. $12a+8b$
29. $-4a-3b$	31. $-17x to the 6$
33. $11y to the 2+4y+11$	35. $-3x to the 2+17x-1$
37. $11x to the 2-5x+5$	39. $6a to the 2-4a-1$
41. $2m to the 2-7m+4$	43. $11a+3$
45. $12s to the 2-14s+9$	47. $3x to the 2-x+4$
49. $w to the 2+3w+4$	51. $2p to the 3+p to the 2+9p+10$
51. $2p to the 3+p to the 2+9p+10$	55. $11w-64$
57. $12c-45$	59. $10x to the 2-7xy+6y to the 2$
61. $10m to the 2+3mn-8n to the 2$	63. $-3ab+3b to the 2$
65. $4uv+2v to the 2$	67. $p to the 3-6p to the 2q+pq to the 2+4q to the 3$
69. $x to the 3+2x to the 2y-5xy to the 2+y to the 3$	71. a) 187 b) 46 c) 2
73. a) −104 b) 4 c) 40	75. 11
77. $10,800	77. $10,800
81. $58	83. 149
85. Answers will vary.	87. Answers will vary.

Attributions

This chapter has been adapted from “Add and Subtract Polynomials” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

8.2 Multiply Polynomials

Learning Objectives

By the end of this section, you will be able to:

Multiply a polynomial by a monomial
Multiply a binomial by a binomial
Multiply a trinomial by a binomial

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like $2(x-3)$ . You multiplied both terms in the parentheses, $x$ and $3$ , by 2, to get $2x-6$ . With this chapter’s new vocabulary, you can say you were multiplying a binomial, $x-3$ , by a monomial, 2

Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

EXAMPLE 1

Multiply: $4(x+3)$ .

Solution


Distribute.
Simplify.

TRY IT 1.1

Multiply: $5(x+7)$ .

Show answer

$5x+35$

TRY IT 1.2

Multiply: $3(y+13)$ .

Show answer

$3y+39$

EXAMPLE 2

Multiply: $y(y-2)$ .

Solution


Distribute.
Simplify.

TRY IT 2.1

Multiply: $x(x-7)$ .

Show answer

$x to the 2-7x$

TRY IT 2.2

Multiply: $d(d-11)$ .

Show answer

$d to the 2-11d$

EXAMPLE 3

Multiply: $7x(2x+y)$ .

Solution


Distribute.
Simplify.

TRY IT 3.1

Multiply: $5x(x+4y)$ .

Show answer

$5x to the 2+20xy$

TRY IT 3.2

Multiply: $2p(6p+r)$ .

Show answer

$12p to the 2+2pr$

EXAMPLE 4

Multiply: $-2y(4y to the 2+3y-5)$ .

Solution


Distribute.
Simplify.

TRY IT 4.1

Multiply: $-3y(5y to the 2+8y-7)$ .

Show answer

$-15y to the 3-24y to the 2+21y$

TRY IT 4.2

Multiply: $4x to the 2(2x to the 2-3x+5)$ .

Show answer

$8x to the 4-24x to the 3+20x to the 2$

EXAMPLE 5

Multiply: $2x to the 3(x to the 2-8x+1)$ .

Solution


Distribute.
Simplify.

TRY IT 5.1

Multiply: $4x(3x to the 2-5x+3)$ .

Show answer

$12x to the 3-20x to the 2+12x$

TRY IT 5.2

Multiply: $-6a to the 3(3a to the 2-2a+6)$ .

Show answer

$-18a to the 5 divided by 12a to the 4-36a to the 3$

EXAMPLE 6

Multiply: $(x+3)p$ .

Solution

The monomial is the second factor.
Distribute.
Simplify.

TRY IT 6.1

Multiply: $(x+8)p$ .

Show answer

$xp+8p$

TRY IT 6.2

Multiply: $(a+4)p$ .

Show answer

$ap+4p$

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

Multiply a Binomial by a Binomial Using the Distributive Property

Look at the table below, where we multiplied a binomial by a monomial.

x plus 3, in parentheses, times p. Two arrows extend from the p, terminating at x and 3.

We distributed the p to get:

What if we have (x + 7) instead of p?

x plus 3 multiplied by x plus 7. Two arrows extend from x plus 7, terminating at the x and the 3 in the first binomial.

Distribute (x + 7).

The sum of two products. The product of x and x plus 7, plus the product of 3 and x plus 7.

Distribute again.

Combine like terms.

Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

EXAMPLE 7

Multiply: $(y+5)(y+8)$ .

Solution


Distribute (y + 8).
Distribute again
Combine like terms.

TRY IT 7.1

Multiply: $(x+8)(x+9)$ .

Show answer

$x to the 2+17x+72$

TRY IT 7.2

Multiply: $(5x+9)(4x+3)$ .

Show answer

$20x to the 2+51x+27$

EXAMPLE 8

Multiply: $(2y+5)(3y+4)$ .

Solution


Distribute (3y + 4).
Distribute again
Combine like terms.

TRY IT 8.1

Multiply: $(3b+5)(4b+6)$ .

Show answer

$12b to the 2+38b+30$

TRY IT 8.2

Multiply: $(a+10)(a+7)$ .

Show answer

$a to the 2+17a+70$

EXAMPLE 9

Multiply: $(4y+3)(2y-5)$ .

Solution


Distribute.
Distribute again.
Combine like terms.

TRY IT 9.1

Multiply: $(5y+2)(6y-3)$ .

Show answer

$30y to the 2-3y-6$

TRY IT 9.2

Multiply: $(3c+4)(5c-2)$ .

Show answer

$15c to the 2+14c-8$

EXAMPLE 10

Multiply: $(x+2)(x-y)$ .

Solution


Distribute.
Distribute again.
There are no like terms to combine.

TRY IT 10.1

Multiply: $(a+7)(a-b)$ .

Show answer

$a to the 2-ab+7a-7b$

TRY IT 10.2

Multiply: $(x+5)(x-y)$ .

Show answer

$x to the 2-xy+5x-5y$

Multiply a Binomial by a Binomial Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes, like in the above example, there are no like terms to combine.

Let’s look at the last example again and pay particular attention to how we got the four terms.

$mathematical expression$

Where did the first term, $x to the 2$ , come from?

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products.

$(x-2)(x-y)$

$mathematical expression$

Let’s look at $(x+3)(x+7)$ .

Distibutive Property

FOIL

The product of x plus 3 and x plus y. An arrow extends from the x in x plus 3 to the x in x plus 7. A second arrow extends from the x in x plus 3 to the 7 in x plus 7. A third arrow extends from the 3 in x plus 3 to the x in x plus 7. A fourth arrow extends from the 3 in x plus 3 to the 7 in x plus 7.

The sum of two products, the product of x and x plus 7, and the product of 3 and x plus 7.

x squared plus 7 x plus 3 x plus 21. Below x squared is the letter F, below 7 x is the letter O, below 3 x is the letter I, and below 21 is the letter L, spelling FOIL.

Notice how the terms in third line fit the FOIL pattern.

Now we will do an example where we use the FOIL pattern to multiply two binomials.

EXAMPLE 11

How to Multiply a Binomial by a Binomial using the FOIL Method

Multiply using the FOIL method: $(x+5)(x+9)$ .

Solution

In the second row, the first cell reads “Step 2. Multiply the outer terms.” In the second cell is the product of x plus 5 and x plus 9 again, with an arrow extending from x in the first binomial to the 9 in the second binomial. The third cell contains x squared plus 9x plus blank plus blank, with the letter F under the x squared, O under the 9x, and I and L beneath the two blanks. In the third row, the first cell reads “Step 3. Multiply the inner terms.” The second cell contains the product of x plus 5 and x plus 9 again, with an arrow extending from 5 in the first binomial to the x in the second binomial. The third cell contains x squared plus 9x plus 5x plus blank, with F beneath x squared, O beneath 9x, I beneath 5x, and L beneath the blank. In the fourth row, the first cell reads “Step 4. Multiply the last terms.” In the second cell is the product of x plus 5 and x plus 9 again, with an arrow extending from 5 in the first binomial to 9 in the second binomial. The third cell contains x squared plus 9x plus 6x plus 45, with F beneath x squared, O beneath 9x, I beneath 6x, and L beneath 45. In the final row, the first cell reads “Step 5. Combine like terms, when possible.” The second cell is blank. The third cell contains the final expression: x squared plus 15x plus 45.

TRY IT 11.1

Multiply using the FOIL method: $(x+6)(x+8)$ .

Show answer

$x to the 2+14x+48$

TRY IT 11.2

Multiply using the FOIL method: $(y+17)(y+3)$ .

Show answer

$y to the 2+20y+51$

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

HOW TO: Multiply two binomials using the FOIL method

When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.

EXAMPLE 12

Multiply: $(y-7)(y+4)$ .

Solution

TRY IT 12.1

Multiply: $(x-7)(x+5)$ .

Show answer

$x to the 2-2x-35$

TRY IT 12.2

Multiply: $(b-3)(b+6)$ .

Show answer

$b to the 2+3b-18$

EXAMPLE 13

Multiply: $(4x+3)(2x-5)$ .

Solution

TRY IT 13.1

Multiply: $(3x+7)(5x-2)$ .

Show answer

$15x to the 2+29x-14$

TRY IT 13.2

Multiply: $(4y+5)(4y-10)$ .

Show answer

$16y to the 2-20y-50$

The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.

EXAMPLE 14

Multiply: $(3x-y)(2x-5)$ .

Solution



Multiply the First.
Multiply the Outer.
Multiply the Inner.
Multiply the Last.
Combine like terms—there are none.

TRY IT 14.1

Multiply: $(10c-d)(c-6)$ .

Show answer

$10c to the 2-60c-cd+6d$

TRY IT 14.2

Multiply: $(7x-y)(2x-5)$ .

Show answer

$14x to the 2-35x-2xy+10y$

Be careful of the exponents in the next example.

EXAMPLE 15

Multiply: $(n to the 2+4)(n-1)$ .

Solution



Multiply the First.
Multiply the Outer.
Multiply the Inner.
Multiply the Last.
Combine like terms—there are none.

TRY IT 15.1

Multiply: $(x to the 2+6)(x-8)$ .

Show answer

$x to the 3-8x to the 2+6x-48$

TRY IT 15.2

Multiply: $(y to the 2+7)(y-9)$ .

Show answer

$y to the 3-9y to the 2+7y-63$

EXAMPLE 16

Multiply: $(3pq+5)(6pq-11)$ .

Solution


Multiply the First.
Multiply the Outer.
Multiply the Inner.
Multiply the Last.
Combine like terms—there are none.

TRY IT 16.1

Multiply: $(2ab+5)(4ab-4)$ .

Show answer

$8a to the 2b to the 2+12ab-20$

TRY IT 16.2

Multiply: $(2xy+3)(4xy-5)$ .

Show answer

$8x to the 2y to the 2+2xy-15$

Multiply a Binomial by a Binomial Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

Now we’ll apply this same method to multiply two binomials.

EXAMPLE 17

Multiply using the Vertical Method: $(3y-1)(2y-6)$ .

Solution

It does not matter which binomial goes on the top.

Multiply $3y-1$ by $-6$ Partial Product $-18y+6$

This figure has two columns. In the left column is the product of two binomials, 3y minus 1 and 2y minus 6. Below this is 6y squared minus 2y minus 18y plus 6. Below this is 6y squared minus 20y plus 6. In the right column is the vertical multiplication of 3y minus 1 and 2y minus 6. Below this is the partial product negative 18y plus 6. Below this is the partial product 6y squared minus 2y. Below this is 6y squared minus 20y plus 6.

Multiple $3y-1$ by $2y$ Partial Product $6y to the 2 -2y$

Add like terms. Product $6y to the 2 - 20y+6$

Notice the partial products are the same as the terms in the FOIL method.

TRY IT 17.1

Multiply using the Vertical Method: $(5m-7)(3m-6)$ .

Show answer

$15m to the 2-51m+42$

TRY IT 17.2

Multiply using the Vertical Method: $(6b-5)(7b-3)$ .

Show answer

$42b to the 2-53b+15$

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.

HOW TO: Multiplying Two Binomials

To multiply binomials, use the: To multiply binomials, use the:

- Distributive Property
- FOIL Method
- Vertical Method

Remember, FOIL only works when multiplying two binomials.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

EXAMPLE 18

Multiply using the Distributive Property: $(b+3)(2b to the 2-5b+8)$ .

Solution


Distribute.
Multiply.
Combine like terms.

TRY IT 18.1

Multiply using the Distributive Property: $(y-3)(y to the 2-5y+2)$ .

Show answer

$y to the 3-8y to the 2+17y-6$

TRY IT 18.2

Multiply using the Distributive Property: $(x+4)(2x to the 2-3x+5)$ .

Show answer

$2x to the 3+5x to the 2-7x+20$

Now let’s do this same multiplication using the Vertical Method.

EXAMPLE 19

Multiply using the Vertical Method: $(b+3)(2b to the 2-5b+8)$ .

Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Multiply (2b² − 5b + 8) by 3.

Multiply (2b² − 5b + 8) by b.
Add like terms.

TRY IT 19.1

Multiply using the Vertical Method: $(y-3)(y to the 2-5y+2)$ .

Show answer

$y to the 3-8y to the 2+17y-6$

TRY IT 19.2

Multiply using the Vertical Method: $(x+4)(2x to the 2-3x+5)$ .

Show answer

$2x to the 3+5x to the 2-7x+20$

We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

HOW TO: Multiply a Trinomial by a Binomial

To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Access these online resources for additional instruction and practice with multiplying polynomials:

Key Concepts

FOIL Method for Multiplying Two Binomials—To multiply two binomials:
1. Multiply the First terms.
2. Multiply the Outer terms.
3. Multiply the Inner terms.
4. Multiply the Last terms.
Multiplying Two Binomials—To multiply binomials, use the:
- Distributive Property ((Figure))
- FOIL Method ((Figure))
Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:
- Distributive Property ((Figure))

Practice Makes Perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

1. $4(w+10)$	2. $6(b+8)$
3. $-3(a+7)$	4. $-5(p+9)$
5. $2(x-7)$	6. $7(y-4)$
7. $-3(k-4)$	8. $-8(j-5)$
9. $q(q+5)$	10. $k(k+7)$
11. $-b(b+9)$	12. $-y(y+3)$
13. $-x(x-10)$	14. $-p(p-15)$
15. $6r(4r+s)$	16. $5c(9c+d)$
17. $12x(x-10)$	18. $9m(m-11)$
19. $-9a(3a+5)$	20. $-4p(2p+7)$
21. $3(p to the 2+10p+25)$	22. $6(y to the 2+8y+16)$
23. $-8x(x to the 2+2x-15)$	24. $-5t(t to the 2+3t-18)$
25. $5q to the 3(q to the 3-2q+6)$	26. $4x to the 3(x to the 4-3x+7)$
27. $-8y(y to the 2+2y-15)$	28. $-5m(m to the 2+3m-18)$
29. $5q to the 3(q to the 2-2q+6)$	30. $9r to the 3(r to the 2-3r+5)$
31. $-4z to the 2(3z to the 2+12z-1)$	32. $-3x to the 2(7x to the 2+10x-1)$
33. $(2m-9)m$	34. $(8j-1)j$
35. $(w-6) times 8$	36. $(k-4) times 5$
37. $4(x+10)$	38. $6(y+8)$
39. $15(r-24)$	40. $12(v-30)$
41. $-3(m+11)$	42. $-4(p+15)$
43. $-8(z-5)$	44. $-3(x-9)$
45. $u(u+5)$	46. $q(q+7)$
47. $n(n to the 2-3n)$	48. $s(s to the 2-6s)$
49. $6x(4x+y)$	50. $5a(9a+b)$
51. $5p(11p-5q)$	52. $12u(3u-4v)$
53. $3(v to the 2+10v+25)$	54. $6(x to the 2+8x+16)$
55. $2n(4n to the 2-4n+1)$	56. $3r(2r to the 2-6r+2)$
57. $-8y(y to the 2+2y-15)$	58. $-5m(m to the 2+3m-18)$
59. $5q to the 3(q to the 2-2q+6)$	60. $9r to the 3(r to the 2-3r+5)$
61. $-4z to the 2(3z to the 2+12z-1)$	62. $-3x to the 2(7x to the 2+10x-1)$
63. $(2y-9)y$	64. $(8b-1)b$

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: a) the Distributive Property b) the FOIL method c) the Vertical Method.

65. $(w+5)(w+7)$	66. $(y+9)(y+3)$
67. $(p+11)(p-4)$	68. $(q+4)(q-8)$

In the following exercises, multiply the binomials. Use any method.

69. $(x+8)(x+3)$	70. $(y+7)(y+4)$
71. $(y-6)(y-2)$	72. $(x-7)(x-2)$
73. $(w-4)(w+7)$	74. $(q-5)(q+8)$
75. $(p+12)(p-5)$	76. $(m+11)(m-4)$
77. $(6p+5)(p+1)$	78. $(7m+1)(m+3)$
79. $(2t-9)(10t+1)$	80. $(3r-8)(11r+1)$
81. $(5x-y)(3x-6)$	82. $(10a-b)(3a-4)$
83. $(a+b)(2a+3b)$	84. $(r+s)(3r+2s)$
85. $(4z-y)(z-6)$	86. $(5x-y)(x-4)$
87. $(x to the 2+3)(x+2)$	88. $(y to the 2-4)(y+3)$
89. $(x to the 2+8)(x to the 2-5)$	90. $(y to the 2-7)(y to the 2-4)$
91. $(5ab-1)(2ab+3)$	92. $(2xy+3)(3xy+2)$
93. $(6pq-3)(4pq-5)$	94. $(3rs-7)(3rs-4)$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using a) the Distributive Property b) the Vertical Method.

95. $(x+5)(x to the 2+4x+3)$	96. $(u+4)(u to the 2+3u+2)$
97. $(y+8)(4y to the 2+y-7)$	98. $(a+10)(3a to the 2+a-5)$

In the following exercises, multiply. Use either method.

99. $(w-7)(w to the 2-9w+10)$	100. $(p-4)(p to the 2-6p+9)$
101. $(3q+1)(q to the 2-4q-5)$	102. $(6r+1)(r to the 2-7r-9)$

Mixed Practice

103. $(10y-6)+(4y-7)$	104. $(15p-4)+(3p-5)$
105. $(x to the 2-4x-34)-(x to the 2+7x-6)$	106. $(j to the 2-8j-27)-(j to the 2+2j-12)$
107. $5q(3q to the 2-6q+11)$	108. $8t(2t to the 2-5t+6)$
109. $(s-7)(s+9)$	110. $(x-5)(x+13)$
111. $(y to the 2-2y)(y+1)$	112. $(a to the 2-3a)(4a+5)$
113. $(3n-4)(n to the 2+n-7)$	114. $(6k-1)(k to the 2+2k-4)$
115. $(7p+10)(7p-10)$	116. $(3y+8)(3y-8)$
117. $(4m to the 2-3m-7)m to the 2$	118. $(15c to the 2-4c+5)c to the 4$
119. $(5a+7b)(5a+7b)$	120. $(3x-11y)(3x-11y)$
121. $(4y+12z)(4y-12z)$

Everyday Math

122. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15. Think of 13 as $10+3$ and 15 as $10+5$ .

Multiply $(10+3)(10+5)$ by the FOIL method.
Multiply $13 times 15$ without using a calculator.
Which way is easier for you? Why?

123. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as $20-2$ and 17 as $20-3$ .

Multiply $(20-2)(20-3)$ by the FOIL method.
Multiply $18 times 17$ without using a calculator.
Which way is easier for you? Why?

Writing Exercises

124. Which method do you prefer to use when multiplying two binomials: the Distributive Property, the FOIL method, or the Vertical Method? Why?

125. Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?

126. Multiply the following:

$mathematical expression$

Explain the pattern that you see in your answers.

127. Multiply the following:

$mathematical expression$

Explain the pattern that you see in your answers.

128. Multiply the following:

$mathematical expression$

Explain the pattern that you see in your answers.

129. Multiply the following:

$mathematical expression$

Explain the pattern that you see in your answers.

Answers

1. $4w+40$	3. $-3a-21$
5. $2x-14$	7. $-3k+12$
9. $q to the 2+5q$	11. $-b to the 2-9b$
13. $-x to the 2+10x$	15. $24r to the 2+6rs$
17. $12x to the 2-120x$	19. $-27a to the 2-45a$
21. $3p to the 2+30p+75$	23. $-8x to the 3-16x to the 2+120x$
25. $5q to the 6-10q to the 4+30q to the 3$	27. $-8y to the 3-16y to the 2+120y$
29. $5q to the 5-10q to the 4+30q to the 3$	31. $-12z to the 4-48z to the 3+4z to the 2$
33. $2m to the 2-9m$	35. $8w-48$
37. $4x+40$	39. $15r-360$
41. $-3m-33$	43. $-8z+40$
45. $u to the 2+5u$	47. $n to the 3-3n to the 2$
49. $24x to the 2+6xy$	51. $55p to the 2-25pq$
53. $3v to the 2+30v+75$	55. $8n to the 3-8n to the 2+2n$
57. $-8y to the 3-16y to the 2+120y$	59. $5q to the 5-10q to the 4+30q to the 3$
61. $-12z to the 4-48z to the 3+4z to the 2$	63. $2y to the 2-9y$
65. $w to the 2+12w+35$	67. $p to the 2+7p-44$
69. $x to the 2+11x+24$	71. $y to the 2-8y+12$
73. $w to the 2+3w-28$	75. $p to the 2+7p-60$
77. $6p to the 2+11p+5$	79. $20t to the 2-88t-9$
81. $15x to the 2-3xy-30x+6y$	83. $2a to the 2+5ab+3b to the 2$
85. $4z to the 2-24z-zy+6y$	87. $x to the 3+2x to the 2+3x+6$
89. $x to the 4+3x to the 2-40$	91. $10a to the 2b to the 2+13ab-3$
93. $24p to the 2q to the 2-42pq+15$	95. $x to the 3+9x to the 2+23x+15$
97. $4y to the 3+33y to the 2+y-56$	99. $w to the 3-16w to the 2+73w-70$
101. $3q to the 3-11q to the 2-19q-5$	103. $14y-13$
105. $-11x-28$	107. $15q to the 3-30q to the 2+55q$
109. $s to the 2+2s-63$	111. $y to the 3-y to the 2-2y$
113. $3n to the 3-n to the 2-25n+28$	115. $49p to the 2-100$
117. $4m to the 4-3m to the 3-7m to the 2$	119. $25a to the 2+70ab+49b to the 2$
121. $16y to the 2-144z to the 2$	123. a) 306 b) 306 c) Answers will vary.
125. Answers will vary.	127. Answers will vary.
129. Answers will vary.

Attributions

This chapter has been adapted from “Multiply Polynomials” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

8.3 Special Products

Learning Objectives

By the end of this section, you will be able to:

Square a binomial using the Binomial Squares Pattern
Multiply conjugates using the Product of Conjugates Pattern
Recognize and use the appropriate special product pattern

Square a Binomial Using the Binomial Squares Pattern

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and using the methods of the last section, there is less work to do if you learn to use a pattern.

Let’s start by looking at $(x+9) to the 2$ .
What does this mean?	$(x+9) to the 2$
It means to multiply $(x+9)$ by itself.	$(x+9)(x+9)$
Then, using FOIL, we get:	$x to the 2+9x+9x+81$
Combining like terms gives:	$x to the 2+18x+81$

Here’s another one:	$(y-7) to the 2$
Multiply $(y-7)$ by itself.	$(y-7)(y-7)$
Using FOIL, we get:	$y to the 2-7y-7y+49$
And combining like terms:	$y to the 2-14y+49$

And one more:	$(2x+3) to the 2$
Multiply.	$(2x+3)(2x+3)$
Use FOIL:	$4x to the 2+6x+6x+9$
Combine like terms.	$4x to the 2+12x+9$

Look at these results. Do you see any patterns?

What about the number of terms? In each example we squared a binomial and the result was a trinomial.

$(a+b) to the 2$ = ____ + ____ + ____

Now look at the first term in each result. Where did it come from?

The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!

$(a+b) to the 2=a to the 2$ + ____ + ____

To get the first term of the product, square the first term.

Where did the last term come from? Look at the examples and find the pattern.

The last term is the product of the last terms, which is the square of the last term.

$mathematical expression$

To get the last term of the product, square the last term.

Finally, look at the middle term. Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.

$mathematical expression$

To get the middle term of the product, multiply the terms and double their product.

Putting it all together:

Binomial Squares Pattern

If $a$ and $b$ are real numbers,

$mathematical expression$

No Alt Text

HOW TO:

To square a binomial:

square the first term
square the last term
double their product

A number example helps verify the pattern.

$(10+4) to the 2$

Square the first term.

$mathematical expression$

Square the last term.

$mathematical expression$

Double their product.

$10 to the 2+2 times 10 times 4+4 to the 2$

Simplify.

$100+80+16$

Simplify.

$196$

To multiply $(10+4) to the 2$ usually you’d follow the Order of Operations.

$mathematical expression$

The pattern works!

EXAMPLE 1

Multiply: $(x+5) to the 2$ .

Solution


Square the first term.
Square the last term.
Double the product.
Simplify.

TRY IT 1.1

Multiply: $(x+9) to the 2$ .

Show answer

$x to the 2+18x+81$

TRY IT 1.2

Multiply: $(y+11) to the 2$ .

Show answer

$y to the 2+22y+121$

EXAMPLE 2

Multiply: $(y-3) to the 2$ .

Solution

y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.

Square the first term.

y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.

Square the last term.

Double the product.

y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.

Simplify.

TRY IT 2.1

Multiply: $(x-9) to the 2$ .

Show answer

$x to the 2-18x+81$

TRY IT 2.2

Multiply: $(p-13) to the 2$ .

Show answer

$p to the 2-26p+169$

EXAMPLE 3

Multiply: $(4x+6) to the 2$ .

Solution

4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.

Use the pattern.

4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.

Simplify.

TRY IT 3.1

Multiply: $(6x+3) to the 2$ .

Show answer

$36x to the 2+36x+9$

TRY IT 3.2

Multiply: $(4x+9) to the 2$ .

Show answer

$16x to the 2+72x+81$

EXAMPLE 4

Multiply: $(2x-3y) to the 2$ .

Solution

contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.

Use the pattern.

2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.

Simplify.

4 x squared minus 12 x y plus 9 y squared.

TRY IT 4.1

Multiply: $(2c-d) to the 2$ .

Show answer

$4c to the 2-4cd+d to the 2$

TRY IT 4.2

Multiply: $(4x-5y) to the 2$ .

Show answer

$16x to the 2-40xy+25y to the 2$

EXAMPLE 5

Multiply: $(4u to the 3+1) to the 2$ .

Solution

4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.

Use the pattern.

4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.

Simplify.

16 u to the sixth power plus 18 u cubed plus 1.

TRY IT 5.1

Multiply: $(2x to the 2+1) to the 2$ .

Show answer

$4x to the 4+4x to the 2+1$

TRY IT 5.2

Multiply: $(3y to the 3+2) to the 2$ .

Show answer

$9y to the 6+12y to the 3+4$

Multiply Conjugates Using the Product of Conjugates Pattern

We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.

What do you notice about these pairs of binomials?

$mathematical expression$

Look at the first term of each binomial in each pair.

This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses

Notice the first terms are the same in each pair.

Look at the last terms of each binomial in each pair.

Notice the last terms are the same in each pair.

Notice how each pair has one sum and one difference.

A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference has a special name. It is called a conjugate pair and is of the form $(a-b),(a+b)$ .

Conjugate Pair

A conjugate pair is two binomials of the form

$(a-b),(a+b)$ .

The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.

There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier.

Let’s look for the pattern by using FOIL to multiply some conjugate pairs.

$mathematical expression$

Each first term is the product of the first terms of the binomials, and since they are identical it is the square of the first term.

$mathematical expression$

The last term came from multiplying the last terms, the square of the last term.

$mathematical expression$

What do you observe about the products?

The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.

Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.

The product of conjugates is always of the form $a to the 2-b to the 2$ . This is called a difference of squares.

This leads to the pattern:

Product of Conjugates Pattern

If $a$ and $b$ are real numbers,

The product is called a difference of squares.

To multiply conjugates, square the first term, square the last term, and write the product as a difference of squares.

Let’s test this pattern with a numerical example.

	$(10-2)(10+2)$
It is the product of conjudgates, so the result will be the difference of two squares.	____ – ____
Square the first term.	$mathematical expression$
Square the last term.	$10 to the 2-2 to the 2$
Simplify.	$100-4$
Simplify.	$96$
What do you get using the order of operations?
	$mathematical expression$

Notice, the result is the same!

EXAMPLE 6

Multiply: $(x-8)(x+8)$ .

Solution

First, recognize this as a product of conjugates. The binomials have the same first terms, and the same last terms, and one binomial is a sum and the other is a difference.

It fits the pattern.
Square the first term, x.
Square the last term, 8.
The product is a difference of squares.

TRY IT 6.1

Multiply: $(x-5)(x+5)$ .

Show answer

$x to the 2-25$

TRY IT 6.2

Multiply: $(w-3)(w+3)$ .

Show answer

$w to the 2-9$

EXAMPLE 7

Multiply: $(2x+5)(2x-5)$ .

Solution

Are the binomials conjugates?

It is the product of conjugates.
Square the first term, 2x.
Square the last term, 5.
Simplify. The product is a difference of squares.

TRY IT 7.1

Multiply: $(6x+5)(6x-5)$ .

Show answer

$36x to the 2-25$

TRY IT 7.2

Multiply: $(2x+7)(2x-7)$ .

Show answer

$4x to the 2-49$

The binomials in the next example may look backwards – the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.

EXAMPLE 8

Find the product: $(3+5x)(3-5x)$ .

Solution

It is the product of conjugates.
Use the pattern.
Simplify.

TRY IT 8.1

Multiply: $(7+4x)(7-4x)$ .

Show answer

$49-16x to the 2$

TRY IT 8.2

Multiply: $(9-2y)(9+2y)$ .

Show answer

$81-4y to the 2$

Now we’ll multiply conjugates that have two variables.

EXAMPLE 9

Find the product: $(5m-9n)(5m+9n)$ .

Solution

This fits the pattern.
Use the pattern.
Simplify.

TRY IT 9.1

Find the product: $(4p-7q)(4p+7q)$ .

Show answer

$16p to the 2-49q to the 2$

TRY IT 9.2

Find the product: $(3x-y)(3x+y)$ .

Show answer

$9x to the 2-y to the 2$

EXAMPLE 10

Find the product: $(cd-8)(cd+8)$ .

Solution

This fits the pattern.
Use the pattern.
Simplify.

TRY IT 10.1

Find the product: $(xy-6)(xy+6)$ .

Show answer

$x to the 2y to the 2-36$

TRY IT 10.2

Find the product: $(ab-9)(ab+9)$ .

Show answer

$a to the 2b to the 2-81$

EXAMPLE 11

Find the product: $(6u to the 2-11v to the 5)(6u to the 2+11v to the 5)$ .

Solution

This fits the pattern.
Use the pattern.
Simplify.

TRY IT 11.1

Find the product: $(3x to the 2-4y to the 3)(3x to the 2+4y to the 3)$ .

Show answer

$9x to the 4-16y to the 6$

TRY IT 11.2

Find the product: $(2m to the 2-5n to the 3)(2m to the 2+5n to the 3)$ .

Show answer

$4m to the 4-25n to the 6$

Recognize and Use the Appropriate Special Product Pattern

We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

Comparing the Special Product Patterns

Binomial Squares

Product of Conjugates

$(a+b) to the 2=a to the 2+2ab+b to the 2$

$(a-b)(a+b)=a to the 2-b to the 2$

$(a-b) to the 2=a to the 2-2ab+b to the 2$

– Squaring a binomial

– Multiplying conjugates

– Product is a trinomial

– Product is a binomial

– Inner and outer terms with FOIL are the same.

– Inner and outer terms with FOIL are opposites.

– Middle term is double the product of the terms.

– There is no middle term.

EXAMPLE 12

Choose the appropriate pattern and use it to find the product:

a) $(2x-3)(2x+3)$ b) $(5x-8) to the 2$ c) $(6m+7) to the 2$ d) $(5x-6)(6x+5)$

Solution

$(2x-3)(2x+3)$ These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.

This fits the pattern.

Use the pattern.

Simplify.
$(8x-5) to the 2$ We are asked to square a binomial. It fits the binomial squares pattern.

Use the pattern.

Simplify.
$(6m+7) to the 2$ Again, we will square a binomial so we use the binomial squares pattern.

Use the pattern.

Simplify.
$(5x-6)(6x+5)$ This product does not fit the patterns, so we will use FOIL.

$(5x-6)(6x+5)$

Use FOIL. $30x to the 2+25x-36x-30$

Simplify. $30x to the 2-11x-30$

TRY IT 12.1

Choose the appropriate pattern and use it to find the product:

a) $(9b-2)(2b+9)$ b) $(9p-4) to the 2$ c) $(7y+1) to the 2$ d) $(4r-3)(4r+3)$

Show answer

a) FOIL; $18b to the 2+77b-18$ b) Binomial Squares; $81p to the 2-72p+16$ c) Binomial Squares; $49y to the 2+14y+1$ d) Product of Conjugates; $16r to the 2-9$

TRY IT 12.2

Choose the appropriate pattern and use it to find the product:

a) $(6x+7) to the 2$ b) $(3x-4)(3x+4)$ c) $(2x-5)(5x-2)$ d) $(6n-1) to the 2$

Show answer

a) Binomial Squares; $36x to the 2+84x+49$ b) Product of Conjugates; $9x to the 2-16$ c) FOIL; $10x to the 2-29x+10$ d) Binomial Squares; $36n to the 2-12n+1$

Access these online resources for additional instruction and practice with special products:

Special Products

Key Concepts

Binomial Squares Pattern
- If $a,b$ are real numbers,
- $(a+b) to the 2=a to the 2+2ab+b to the 2$
- $(a-b) to the 2=a to the 2-2ab+b to the 2$
- To square a binomial: square the first term, square the last term, double their product.
Product of Conjugates Pattern
- If $a,b$ are real numbers,
- $(a-b)(a+b)=a to the 2-b to the 2$
- The product is called a difference of squares.
To multiply conjugates:
- square the first term square the last term write it as a difference of squares

Glossary

conjugate pair: A conjugate pair is two binomials of the form $(a-b),(a+b)$ ; the pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.

Practice Makes Perfect

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

1. $(q+12) to the 2$	2. $(w+4) to the 2$
3. $(x+2 over 3) to the 2$	4. $(y+1 over 4) to the 2$
5. $(y-6) to the 2$	6. $(b-7) to the 2$
7. $(p-13) to the 2$	8. $(m-15) to the 2$
9. $(4a+10) to the 2$	10. $(3d+1) to the 2$
11. $(3z+1 over 5) to the 2$	12. $(2q+1 over 3) to the 2$
13. $(2y-3z) to the 2$	14. $(3x-y) to the 2$
15. $(1 over 8x-1 over 9y) to the 2$	16. $(1 over 5x-1 over 7y) to the 2$
17. $(5u to the 2+9) to the 2$	18. $(3x to the 2+2) to the 2$
19. $(8p to the 3-3) to the 2$	20. $(4y to the 3-2) to the 2$

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

Multiply Conjugates Using the Product of Conjugates Pattern

21. $(c-5)(c+5)$	22. $(m-7)(m+7)$
23. $(b+6 over 7)(b-6 over 7)$	24. $(x+3 over 4)(x-3 over 4)$
25. $(8j+4)(8j-4)$	26. $(5k+6)(5k-6)$
27. $(9c+5)(9c-5)$	28. $(11k+4)(11k-4)$
29. $(13-q)(13+q)$	30. $(11-b)(11+b)$
31. $(4-6y)(4+6y)$	32. $(5-3x)(5+3x)$
33. $(7w+10x)(7w-10x)$	34. $(9c-2d)(9c+2d)$
35. $(p+4 over 5q)(p-4 over 5q)$	36. $(m+2 over 3n)(m-2 over 3n)$
37. $(xy-9)(xy+9)$	38. $(ab-4)(ab+4)$
39. $(rs-2 over 7)(rs+2 over 7)$	40. $(uv-3 over 5)(uv+3 over 5)$
41. $(6m to the 3-4n to the 5)(6m to the 3+4n to the 5)$	42. $(2x to the 2-3y to the 4)(2x to the 2+3y to the 4)$
43. $(15m to the 2-8n to the 4)(15m to the 2+8n to the 4)$	44. $(12p to the 3-11q to the 2)(12p to the 3+11q to the 2)$

In the following exercises, find each product.

Recognize and Use the Appropriate Special Product Pattern

45.

a) $(2r+12) to the 2$

b) $(3p+8)(3p-8)$

c) $(7a+b)(a-7b)$

d) $(k-6) to the 2$

46.

a) $(p-3)(p+3)$

b) $(t-9) to the 2$

c) $(m+n) to the 2$

d) $(2x+y)(x-2y)$

47.

a) $(x to the 5+y to the 5)(x to the 5-y to the 5)$

b) $(m to the 3-8n) to the 2$

c) $(9p+8q) to the 2$

d) $(r to the 2-s to the 3)(r to the 3+s to the 2)$

48.

a) $(a to the 5-7b) to the 2$

b) $(x to the 2+8y)(8x-y to the 2)$

c) $(r to the 6+s to the 6)(r to the 6-s to the 6)$

d) $(y to the 4+2z) to the 2$

Everyday Math

49. Mental math You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as $60+5$ .

Multiply $(60+5) to the 2$ by using the binomial squares pattern, $(a+b) to the 2=a to the 2+2ab+b to the 2$ .
Square 65 without using a calculator.
Which way is easier for you? Why?

50. Mental math You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as $50-3$ and 53 as $50+3$ .

Multiply $(50-3)(50+3)$ by using the product of conjugates pattern, $(a-b)(a+b)=a to the 2-b to the 2$ .
Multiply $47 times 53$ without using a calculator.
Which way is easier for you? Why?

Writing Exercises

52. Why does $(a+b) to the 2$ result in a trinomial, but $(a-b)(a+b)$ result in a binomial?	51. How do you decide which pattern to use?
54. Use the order of operations to show that $(3+5) to the 2$ is 64, and then use that numerical example to explain why $(a+b) to the 2 not equal to a to the 2+b to the 2$ .	53. Marta did the following work on her homework paper: $mathematical expression$ Explain what is wrong with Marta’s work.

Answers

1. $q to the 2+24q+144$	3. $x to the 2+4 over 3x+4 over 9$
5. $y to the 2-12y+36$	7. $p to the 2-26p+169$
9. $16a to the 2+80a+100$	11. $9z to the 2+6 over 5z+1 over 25$
13. $4y to the 2-12yz+9z to the 2$	15. $1 over 64x to the 2-1 over 36xy+1 over 81y to the 2$
17. $25u to the 4+90u to the 2+81$	19. $64p to the 6-48p to the 3+9$
21. $c to the 2-25$	23. $b to the 2-36 over 49$
25. $64j to the 2-16$	27. $81c to the 2-25$
29. $169-q to the 2$	31. $16-36y to the 2$
33. $49w to the 2-100x to the 2$	35. $p to the 2-16 over 25q to the 2$
37. $x to the 2y to the 2-81$	39. $r to the 2s to the 2-4 over 49$
41. $36m to the 6-16n to the 10$	43. $225m to the 4-64n to the 8$
45. a) $4r to the 2+48r+144$ b) $9p to the 2-64$ c) $7a to the 2-48ab-7b to the 2$ d) $k to the 2-12k+36$	47. a) $x to the 10-y to the 10$ b) $m to the 6-16m to the 3n+64n to the 2$ c) $81p to the 2+144pq+64q to the 2$ d) $r to the 5+r to the 2s to the 2-r to the 3s to the 3-s to the 5$
49. a) 4,225 b) 4,225 c) Answers will vary.	51. Answers will vary.
53. Answers will vary.

Attributions

This chapter has been adapted from “Special Products” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

8.4 Greatest Common Factor and Factor by Grouping

Learning Objectives

By the end of this section, you will be able to:

Find the greatest common factor of two or more expressions
Factor the greatest common factor from a polynomial
Factor by grouping

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

First we’ll find the GCF of two numbers.

EXAMPLE 1

How to Find the Greatest Common Factor of Two or More Expressions

Find the GCF of 54 and 36

Solution

The second row has the second step of “in each column, circle the common factors. The second column in the second row has the statement “circle the 2, 3 and 3 that are shared by both numbers”. The third column in the second row has the prime factors of 36 and 54 in rows above each other. The common factors of 2, 3, and 3 are circled. The third row has the step “bring down the common factors that all expressions share”. The second column in the third row has “bring down the 2,3, and 3 then multiply”. The third column in the third row has “GCF = 2 times 3 times 3”. The fourth row has the fourth step “multiply the factors”. The second column in the fourth row is blank. The third column in the fourth row has “GCF = 18” and “the GCF of 54 and 36 is 18”.

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18

$mathematical expression$

TRY IT 1.1

Find the GCF of 48 and 80.

Show answer

TRY IT 1.2

Find the GCF of 18 and 40.

Show answer

We summarize the steps we use to find the GCF below.

HOW TO:

Find the Greatest Common Factor (GCF) of two expressions

Factor each coefficient into primes. Write all variables with exponents in expanded form.
List all factors—matching common factors in a column. In each column, circle the common factors.
Bring down the common factors that all expressions share.
Multiply the factors.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

EXAMPLE 2

Find the greatest common factor of $27x to the 3$ and $18x to the 4$ .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
	The GCF of $27x to the 3$ and $18x to the 4$ is $9x to the 3$ .

TRY IT 2.1

Find the GCF: $12x to the 2,18x to the 3$ .

Show answer

$3x to the 2$

TRY IT 2.2

Find the GCF: $16y to the 2,24y to the 3$ .

Show answer

$8y to the 2$

EXAMPLE 3

Find the GCF of $4x to the 2y,6xy to the 3$ .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
	The GCF of $4x to the 2y$ and $6xy to the 3$ is $2xy$ .

TRY IT 3.1

Find the GCF: $6ab to the 4,8a to the 2b$ .

Show answer

$2ab$

TRY IT 3.2

Find the GCF: $9m to the 5n to the 2,12m to the 3n$ .

Show answer

$3m to the 3n$

EXAMPLE 4

Find the GCF of: $21x to the 3,9x to the 2,15x$ .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
	The GCF of $21x to the 3$ , $9x to the 2$ and $15x$ is $3x$ .

TRY IT 4.1

Find the greatest common factor: $25m to the 4,35m to the 3,20m to the 2$ .

Show answer

$5m to the 2$

TRY IT 4.2

Find the greatest common factor: $14x to the 3,70x to the 2,105x$ .

Show answer

$7x$

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as $2 times 6$ or $3 times 4)$ , in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

$mathematical expression$

Now we will start with a product, like $2x+14$ , and end with its factors, $2(x+7)$ . To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

Distributive Property

If $a,b,c$ are real numbers, then

$mathematical expression$

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

EXAMPLE 5

How to Factor the Greatest Common Factor from a Polynomial

Factor: $4x+12$ .

Solution

The second row has the second step “rewrite each term as a product using the G C F”. The second column in the second row has the statement “Rewrite 4 x and 12 as products of their G C F, 4” Then the two equations 4 x = 4 times x and 12 = 4 times 3. The third column in the second row has the expressions 4x + 12 and below this 4 times x + 4 times 3. The third row has the step “Use the reverse distributive property to factor the expression”. The second column in the third row is blank. The third column in the third row has “4(x + 3)”. The fourth row has the fourth step “check by multiplying the factors”. The second column in the fourth row is blank. The third column in the fourth row has three expressions. The first is 4(x + 3), the second is 4 times x + 4 times 3. The third is 4 x + 12.

TRY IT 5.1

Factor: $6a+24$ .

Show answer

$6(a+4)$

TRY IT 5.2

Factor: $2b+14$ .

Show answer

$2(b+7)$

HOW TO:

Factor the greatest common factor from a polynomial.

Find the GCF of all the terms of the polynomial.
Rewrite each term as a product using the GCF.
Use the “reverse” Distributive Property to factor the expression.
Check by multiplying the factors.

Factor as a Noun and a Verb

We use “factor” as both a noun and a verb.

This figure has two statements. The first statement has “noun”. Beside it the statement “7 is a factor of 14” labeling the word factor as the noun. The second statement has “verb”. Beside this statement is “factor 3 from 3a + 3 labeling factor as the verb.

EXAMPLE 6

Factor: $5a+5$ .

Solution

Find the GCF of 5a and 5.

Rewrite each term as a product using the GCF.
Use the Distributive Property “in reverse” to factor the GCF.
Check by mulitplying the factors to get the orginal polynomial.
$5(a+1)$
$5 times a+5 times 1$
$mathematical expression$

TRY IT 6.1

Factor: $14x+14$ .

Show answer

$14(x+1)$

TRY IT 6.1

Factor: $12p+12$ .

Show answer

$12(p+1)$

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

EXAMPLE 7

Factor: $12x-60$ .

Solution

Find the GCF of 12x and 60.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying the factors.
$12(x-5)$
$12 times x-12 times 5$
$mathematical expression$

TRY IT 7.1

Factor: $18u-36$ .

Show answer

$8(u-2)$

TRY IT 7.2

Factor: $30y-60$ .

Show answer

$30(y-2)$

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

EXAMPLE 8

Factor: $4y to the 2+24y+28$ .

Solution

We start by finding the GCF of all three terms.

Find the GCF of $4y to the 2$ , $24y$ and 28.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying.
$4(y to the 2+6y+7)$
$4 times y to the 2+4 times 6y+4 times 7$
$mathematical expression$

TRY IT 8.1

Factor: $5x to the 2-25x+15$ .

Show answer

$5(x to the 2-5x+3)$

TRY IT 8.2

Factor: $3y to the 2-12y+27$ .

Show answer

$3(y to the 2-4y+9)$

EXAMPLE 9

Factor: $5x to the 3-25x to the 2$ .

Solution

Find the GCF of $5x to the 3$ and $25x to the 2$ .

Rewrite each term.
Factor the GCF.
Check.
$5x to the 2(x-5)$
$5x to the 2 times x-5x to the 2 times 5$
$mathematical expression$

TRY IT 9.1

Factor: $2x to the 3+12x to the 2$ .

Show answer

$2x to the 2(x+6)$

TRY IT 9.2

Factor: $6y to the 3-15y to the 2$ .

Show answer

$3y to the 2(2y-5)$

EXAMPLE 10

Factor: $21x to the 3-9x to the 2+15x$ .

Solution

In a previous example we found the GCF of $21x to the 3,9x to the 2,15x$ to be $3x$ .

Rewrite each term using the GCF, 3x.

Factor the GCF.

Check.

$3x(7x to the 2-3x+5)$

$3x times 7x to the 2-3x times 3x+3x times 5$

$mathematical expression$

TRY IT 10.1

Factor: $20x to the 3-10x to the 2+14x$ .

Show answer

$2x(10x to the 2-5x+7)$

TRY IT 10.2

Factor: $24y to the 3-12y to the 2-20y$ .

Show answer

$4y(6y to the 2-3y-5)$

EXAMPLE 11

Factor: $8m to the 3-12m to the 2n+20mn to the 2$ .

Solution

Find the GCF of $8m to the 3$ , $12m to the 2n$ , $20mn to the 2$ .

Rewrite each term.
Factor the GCF.
Check.
$4m(2m to the 2-3mn+5n to the 2)$
$4m times 2m to the 2-4m times 3mn+4m times 5n to the 2$
$mathematical expression$

TRY IT 11.1

Factor: $9xy to the 2+6x to the 2y to the 2+21y to the 3$ .

Show answer

$3y to the 2(3x+2x to the 2+7y)$

TRY IT 11.2

Factor: $3p to the 3-6p to the 2q+9pq to the 3$ .

Show answer

$3p(p to the 2-2pq+3q to the 2)$

When the leading coefficient is negative, we factor the negative out as part of the GCF.

EXAMPLE 12

Factor: $-8y-24$ .

Solution

When the leading coefficient is negative, the GCF will be negative.

Ignoring the signs of the terms, we first find the GCF of 8y and 24 is 8. Since the expression −8y − 24 has a negative leading coefficient, we use −8 as the GCF.

Rewrite each term using the GCF.

Factor the GCF.

Check.

$-8(y+3)$

$-8 times y+(-8) times 3$

$mathematical expression$

TRY IT 12.1

Factor: $-16z-64$ .

Show answer

$-8(8z+8)$

TRY IT 12.2

Factor: $-9y-27$ .

Show answer

$-9(y+3)$

EXAMPLE 13

Factor: $-6a to the 2+36a$ .

Solution

The leading coefficient is negative, so the GCF will be negative.?

Since the leading coefficient is negative, the GCF is negative, −6a.

Rewrite each term using the GCF.

Factor the GCF.

Check.

$-6a(a-6)$

$-6a times a+(-6a)(-6)$

$mathematical expression$

TRY IT 13.1

Factor: $-4b to the 2+16b$ .

Show answer

$-4b(b-4)$

TRY IT 13.2

Factor: $-7a to the 2+21a$ .

Show answer

$-7a(a-3)$

EXAMPLE 14

Factor: $5q(q+7)-6(q+7)$ .

Solution

The GCF is the binomial $q+7$ .

Factor the GCF, (q + 7).

Check on your own by multiplying.

TRY IT 14.1

Factor: $4m(m+3)-7(m+3)$ .

Show answer

$(m+3)(4m-7)$

TRY IT 14.2

Factor: $8n(n-4)+5(n-4)$ .

Show answer

$(n-4)(8n+5)$

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

EXAMPLE 15

How to Factor by Grouping

Factor: $xy+3y+2x+6$ .

Solution

This table gives the steps for factoring x y + 3 y + 2 x + 6. In the first row there is the statement, “group terms with common factors”. In the next column, there is the statement of no common factors of all 4 terms. The last column shows the first two terms grouped and the last two terms grouped. The second row has the statement, “factor out the common factor from each group”. The second column in the second row states to factor out the GCF from the two separate groups. The third column in the second row has the expression y(x + 3) + 2(x + 3). The third row has the statement, “factor the common factor from the expression”. The second column in this row points out there is a common factor of (x + 3). The third column in the third row shows the factor of (x + 3) factored from the two groups, (x + 3) times (y + 2). The last row has the statement, “check”. The second column in this row states to multiply (x + 3)(y + 2). The product is shown in the last column of the original polynomial x y + 3 y + 2 x + 6.

TRY IT 15.1

Factor: $xy+8y+3x+24$ .

Show answer

$(x+8)(y+3)$

TRY IT 15.2

Factor: $ab+7b+8a+56$ .

Show answer

$(a+7)(b+8)$

HOW TO:

Factor by grouping.

Group terms with common factors.
Factor out the common factor in each group.
Factor the common factor from the expression.
Check by multiplying the factors.

EXAMPLE 16

Factor: $x to the 2+3x-2x-6$ .

Solution

There is no GCF in all four terms.	$x to the 2+3x-2x-6$
Separate into two parts.	$mathematical expression$
Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms.	$mathematical expression$
Check on your own by multiplying.

TRY IT 16.1

Factor: $x to the 2+2x-5x-10$ .

Show answer

$(x-5)(x+2)$

TRY IT 16.2

Factor: $y to the 2+4y-7y-28$ .

Show answer

$(y+4)(y-7)$

Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

Key Concepts

Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Bring down the common factors that all expressions share.
4. Multiply the factors.
Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
1. Find the GCF of all the terms of the polynomial.
2. Rewrite each term as a product using the GCF.
3. Use the ‘reverse’ Distributive Property to factor the expression.
4. Check by multiplying the factors.
Factor by Grouping: To factor a polynomial with 4 four or more terms
1. Group terms with common factors.
2. Factor out the common factor in each group.
3. Factor the common factor from the expression.
4. Check by multiplying the factors.

Glossary

factoring: Factoring is splitting a product into factors; in other words, it is the reverse process of multiplying.

greatest common factor: The greatest common factor is the largest expression that is a factor of two or more expressions is the greatest common factor (GCF).

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Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

1. 8, 18	2. 24, 40
3. 72, 162	4. 150, 275
5. 10a, 50	6. 5b, 30
7. $3x,10x to the 2$	8. $21b to the 2,14b$
9. $8w to the 2,24w to the 3$	10. $30x to the 2,18x to the 3$
11. $10p to the 3q,12pq to the 2$	12. $8a to the 2b to the 3,10ab to the 2$
13. $12m to the 2n to the 3,30m to the 5n to the 3$	14. $28x to the 2y to the 4,42x to the 4y to the 4$
15. $10a to the 3,12a to the 2,14a$	16. $20y to the 3,28y to the 2,40y$
17. $35x to the 3,10x to the 4,5x to the 5$	18. $27p to the 2,45p to the 3,9p to the 4$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

19. $4x+20$	20. $8y+16$
21. $6m+9$	22. $14p+35$
23. $9q+9$	24. $7r+7$
25. $8m-8$	26. $4n-4$
27. $9n-63$	28. $45b-18$
29. $3x to the 2+6x-9$	30. $4y to the 2+8y-4$
31. $8p to the 2+4p+2$	32. $10q to the 2+14q+20$
33. $8y to the 3+16y to the 2$	34. $12x to the 3-10x$
35. $5x to the 3-15x to the 2+20x$	36. $8m to the 2-40m+16$
37. $12xy to the 2+18x to the 2y to the 2-30y to the 3$	38. $21pq to the 2+35p to the 2q to the 2-28q to the 3$
39. $-2x-4$	40 $-3b+12$
41. $5x(x+1)+3(x+1)$	42. $2x(x-1)+9(x-1)$
43. $3b(b-2)-13(b-2)$	44. $6m(m-5)-7(m-5)$

Factor by Grouping

In the following exercises, factor by grouping.

45. $xy+2y+3x+6$	46. $mn+4n+6m+24$
47. $uv-9u+2v-18$	48. $pq-10p+8q-80$
49. $b to the 2+5b-4b-20$	50. $m to the 2+6m-12m-72$
51. $p to the 2+4p-9p-36$	52. $x to the 2+5x-3x-15$

Mixed Practice

In the following exercises, factor.

53. $-20x-10$	54. $5x to the 3-x to the 2+x$
55. $3x to the 3-7x to the 2+6x-14$	56. $x to the 3+x to the 2-x-1$
57. $x to the 2+xy+5x+5y$	58. $5x to the 3-3x to the 2-5x-3$

Everyday Math

59. Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression $w to the 2-6w$ , where $w=$ width. Factor the greatest common factor from the polynomial.

60. Height of a baseball The height of a baseball t seconds after it is hit is given by the expression $-16t to the 2+80t+4$ . Factor the greatest common factor from the polynomial.

Writing Exercises

61. The greatest common factor of 36 and 60 is 12. Explain what this means.

62. What is the GCF of $y to the 4,y to the 5$ , and $y to the 10$ ? Write a general rule that tells you how to find the GCF of $y to the a,y to the b$ , and $y to the c$ .

Answers

1. 2	3. 18
5. 10	7. $x$
9. $8w to the 2$	11. $2pq$
13. $6m to the 2n to the 3$	15. $2a$
17. $5x to the 3$	19. $4(x+5)$
21. $3(2m+3)$	23. $9(q+1)$
25. $8(m-1)$	27. $9(n-7)$
29. $3(x to the 2+2x-3)$	31. $2(4p to the 2+2p+1)$
33. $8y to the 2(y+2)$	35. $5x(x to the 2-3x+4)$
37. $6y to the 2(2x+3x to the 2-5y)$	39. $-2(x+4)$
41. $(x+1)(5x+3)$	43. $(b-2)(3b-13)$
45. $(y+3)(x+2)$	47. $(u+2)(v-9)$
49. $(b-4)(b+5)$	51. $(p-9)(p+4)$
53. $-10(2x+1)$	55. $(x to the 2+2)(3x-7)$
57. $(x+y)(x+5)$	59. $w(w-6)$
61. Answers will vary.

Attributions

This chapter has been adapted from “Greatest Common Factor and Factor by Grouping” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

8.5 Factor Quadratic Trinomials with Leading Coefficient 1

Learning Objectives

By the end of this section, you will be able to:

Factor trinomials of the form $x to the 2+bx+c$
Factor trinomials of the form $x to the 2+bxy+cy to the 2$

Factor Trinomials of the Form x² + bx + c

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.

This figure shows the steps of multiplying the factors (x + 2) times (x + 3). The multiplying is completed using FOIL to demonstrate. The first term is x squared and is below F. The second term is 3 x below “O”. The third term is 2 x below “I”. The fourth term is 6 below L. The simplified product is then given as x 2 plus 5 x + 6.

To factor the trinomial means to start with the product, $x to the 2+5x+6$ , and end with the factors, $(x+2)(x+3)$ . You need to think about where each of the terms in the trinomial came from.

The first term came from multiplying the first term in each binomial. So to get $x to the 2$ in the product, each binomial must start with an x.

$mathematical expression$

The last term in the trinomial came from multiplying the last term in each binomial. So the last terms must multiply to 6

What two numbers multiply to 6?

The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use?

Consider the middle term. It came from adding the outer and inner terms.

So the numbers that must have a product of 6 will need a sum of 5. We’ll test both possibilities and summarize the results in the table below—the table will be very helpful when you work with numbers that can be factored in many different ways.

Factors of $6$

Sum of factors

$1,6$

$1+6=7$

$2,3$

$2+3=5$

We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of $x to the 2+5x+6$ . They are $(x+2)(x+3)$ .

$mathematical expression$

You should check this by multiplying.

Looking back, we started with $x to the 2+5x+6$ , which is of the form $x to the 2+bx+c$ , where $b=5$ and $c=6$ . We factored it into two binomials of the form $(x+m)$ and $(x+n)$ .

$mathematical expression$

To get the correct factors, we found two numbers m and n whose product is c and sum is b.

EXAMPLE 1

How to Factor Trinomials of the Form $x to the 2+bx+c$

Factor: $x to the 2+7x+12$ .

Solution

The second row states the second step “find two numbers m and n that multiply to c, m times n = c and add to b, m + n = b”. In the second column of the second row are the factors of 12 and their sums. 1,12 with sum 1 + 12 = 13. 2, 6 with sum 2 + 6 =8. 3, 4 with sum 3 + 4 = 7. The third row states “use m and n as the last terms of the factors”. The second column states “use 3 and 4 as the last terms of the binomials”. The third column in this row has the product (x + 3)(x + 4). In the fourth row the statement is “check by multiplying the factors”. The product of (x + 3)(x +4) is shown to be x 2 + 7 x + 12.

TRY IT 1.1

Factor: $x to the 2+6x+8$ .

Show answer

$(x+2)(x+4)$

TRY IT 1.2

Factor: $y to the 2+8y+15$ .

Show answer

$(y+3)(y+5)$

Let’s summarize the steps we used to find the factors.

HOW TO:

Factor trinomials of the form $x to the 2+bx+c$ .

Write the factors as two binomials with first terms x: $(x)(x)$ .
Find two numbers m and n that
Multiply to c, $m times n=c$
Add to b, $m+n=b$
Use m and n as the last terms of the factors: $(x+m)(x+n)$ .
Check by multiplying the factors.

EXAMPLE 2

Factor: $u to the 2+11u+24$ .

Solution

Notice that the variable is u, so the factors will have first terms u.

$mathematical expression$

Find two numbers that: multiply to 24 and add to 11

Factors of $24$	Sum of factors
$1,24$	$1+24=25$
$2,12$	$2+12=14$
$3,8$	$3+8=11*$
$4,6$	$4+6=10$

Use 3 and 8 as the last terms of the binomials. $(u+3)(u+8)$

Check.

$mathematical expression$

TRY IT 2.1

Factor: $q to the 2+10q+24$ .

Show answer

$(q+4)(q+6)$

TRY IT 2.2

Factor: $t to the 2+14t+24$ .

Show answer

$(t+2)(t+12)$

EXAMPLE 3

Factor: $y to the 2+17y+60$ .

Solution

$mathematical expression$

Find two numbers that multiply to 60 and add to 17

Factors of $60$	Sum of factors
$1,60$	$1+60=61$
$2,30$	$2+30=32$
$3,20$	$3+20=23$
$4,15$	$4+15=19$
$5,12$	$5+12=17*$
$6,10$	$6+10=16$

Use 5 and 12 as the last terms. $(y+5)(y+12)$

Check.

$mathematical expression$

TRY IT 3.1

Factor: $x to the 2+19x+60$ .

Show answer

$(x+4)(x+15)$

TRY IT 3.2

Factor: $v to the 2+23v+60$ .

Show answer

$(v+3)(v+20)$

Factor Trinomials of the Form x² + bx + c with b Negative, c Positive

In the examples so far, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.

Remember: To get a negative sum and a positive product, the numbers must both be negative.

Again, think about FOIL and where each term in the trinomial came from. Just as before,

the first term, $x to the 2$ , comes from the product of the two first terms in each binomial factor, x and y;
the positive last term is the product of the two last terms
the negative middle term is the sum of the outer and inner terms.

How do you get a positive product and a negative sum? With two negative numbers.

EXAMPLE 4

Factor: $t to the 2-11t+28$ .

Solution

Again, with the positive last term, 28, and the negative middle term, $-11t$ , we need two negative factors. Find two numbers that multiply 28 and add to $-11$ .

$mathematical expression$

Find two numbers that: multiply to 28 and add to $-11$ .

Factors of $28$	Sum of factors
$-1,-28$	$-1+(-28)=-29$
$-2,-14$	$-2+(-14)=-16$
$-4,-7$	$-4+(-7)=-11*$

Use -4, -7 as the last terms of the binomials. $(t-4)(t-7)$

Check.

$mathematical expression$

TRY IT 4.1

Factor: $u to the 2-9u+18$ .

Show answer

$(u-3)(u-6)$

TRY IT 4.2

Factor: $y to the 2-16y+63$ .

Show answer

$(y-7)(y-9)$

Factor Trinomials of the Form x² + bx + c with c Negative

Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.

Remember: To get a negative product, the numbers must have different signs.

EXAMPLE 5

Factor: $z to the 2+4z-5$ .

Solution

To get a negative last term, multiply one positive and one negative. We need factors of $-5$ that add to positive 4

Factors of $-5$	Sum of factors
$1,-5$	$1+(-5)=-4$
$-1,5$	$-1+5=4*$

Notice: We listed both $1,-5$ and $-1,5$ to make sure we got the sign of the middle term correct.

$mathematical expression$

Check.

$mathematical expression$

TRY IT 5.1

Factor: $h to the 2+4h-12$ .

Show answer

$(h-2)(h+6)$

TRY IT 5.2

Factor: $k to the 2+k-20$ .

Show answer

$(k-4)(k+5)$

Let’s make a minor change to the last trinomial and see what effect it has on the factors.

EXAMPLE 6

Factor: $z to the 2-4z-5$ .

Solution

This time, we need factors of $-5$ that add to $-4$ .

Factors of $-5$	Sum of factors
$1,-5$	$1+(-5)=-4*$
$-1,5$	$-1+5=4$

$mathematical expression$

Check.

$mathematical expression$

Notice that the factors of $z to the 2-4z-5$ are very similar to the factors of $z to the 2+4z-5$ . It is very important to make sure you choose the factor pair that results in the correct sign of the middle term.

TRY IT 6.1

Factor: $x to the 2-4x-12$ .

Show answer

$(x+2)(x-6)$

TRY IT 6.2

Factor: $y to the 2-y-20$ .

Show answer

$(y+4)(y-5)$

EXAMPLE 7

Factor: $q to the 2-2q-15$ .

Solution

$mathematical expression$

Factors of $-15$	Sum of factors
$1,-15$	$1+(-15)=-14$
$-1,15$	$-1+15=14$
$3,-5$	$3+(-5)=-2*$
$-3,5$	$-3+5=2$

Check.

$(q+3)(q-5)$

$q to the 2-5q+3q-15$

$mathematical expression$

TRY IT 7.1

Factor: $r to the 2-3r-40$ .

Show answer

$(r+5)(r-8)$

TRY IT 7.1

Factor: $s to the 2-3s-10$ .

Show answer

$(s+2)(s-5)$

Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.

EXAMPLE 8

Factor: $y to the 2-6y+15$ .

Solution

$mathematical expression$

Factors of 15	Sum of factors
$-1,-15$	$-1+(-15)=-16$
$-3,-5$	$mathematical expression$

As shown in the table, none of the factors add to $-6$ ; therefore, the expression is prime.

TRY IT 8.1

Factor: $m to the 2+4m+18$ .

Show answer

prime

TRY IT 8.2

Factor: $n to the 2-10n+12$ .

Show answer

prime

EXAMPLE 9

Factor: $2x+x to the 2-48$ .

Solution

$mathematical expression$

As shown in the table, you can use $-6,8$ as the last terms of the binomials.

$(x-6)(x+8)$

Factors of $-48$	Sum of factors
$-1,48$	$-1+48=47$
$-2,24$ $-3,16$ $-4,12$ $-6,8$	$-2+24=22$ $-3+16=13$ $-4+12=8$ $mathematical expression$

Check.

$(x-6)(x+8)$

$x to the 2-6q+8q-48$

$mathematical expression$

TRY IT 9.1

Factor: $9m+m to the 2+18$ .

Show answer

$(m+3)(m+6)$

TRY IT 9.2

Factor: $-7n+12+n to the 2$ .

Show answer

$(n-3)(n-4)$

Let’s summarize the method we just developed to factor trinomials of the form $x to the 2+bx+c$ .

HOW TO:

Factor trinomials of the form $x to the 2+bx+c$ .

When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

$mathematical expression$

When c is positive, m and n have the same sign.

$mathematical expression$

When c is negative, m and n have opposite signs.

$mathematical expression$

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.

Factor Trinomials of the Form x² + bxy + cy²

Sometimes you’ll need to factor trinomials of the form $x to the 2+bxy+cy to the 2$ with two variables, such as $x to the 2+12xy+36y to the 2$ . The first term, $x to the 2$ , is the product of the first terms of the binomial factors, $x times x$ . The $y to the 2$ in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in the previous objective.

EXAMPLE 10

Factor: $x to the 2+12xy+36y to the 2$ .

Solution

$mathematical expression$

Find the numbers that multiply to 36 and add to 12

Factors of $36$	Sum of factors
1, 36	$1+36=37$
2, 18	$2+18=20$
3, 12	$3+12=15$
4, 9	$4+9=13$
6, 6	$6+6=12*$

Use 6 and 6 as the coefficients of the last terms. $(x+6y)(x+6y)$

Check your answer.

$mathematical expression$

TRY IT 10.1

Factor: $u to the 2+11uv+28v to the 2$ .

Show answer

$(u+4v)(u+7v)$

TRY IT 10.2

Factor: $x to the 2+13xy+42y to the 2$ .

Show answer

$(x+6y)(x+7y)$

EXAMPLE 11

Factor: $r to the 2-8rx-9s to the 2$ .

Solution

We need $r$ in the first term of each binomial and $s$ in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

$mathematical expression$

Find the numbers that multiply to $-9$ and add to $-8$ .

Factors of $-9$	Sum of factors
$1,-9$	$1+(-9)=-8*$
$-1,9$	$-1+9=8$
$3,-3$	$3+(-3)=0$

Check your answer.Use 1, -9 as coefficients of the last terms. $(r+s)(r-9s)$

$mathematical expression$

TRY IT 11.1

Factor: $a to the 2-11ab+10b to the 2$ .

Show answer

$(a-b)(a-10b)$

TRY IT 11.2

Factor: $m to the 2-13mn+12n to the 2$ .

Show answer

$(m-n)(m-12n)$

EXAMPLE 12

Factor: $u to the 2-9uv-12v to the 2$ .

Solution

We need u in the first term of each binomial and $v$ in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

$mathematical expression$

Find the numbers that multiply to $-12$ and add to $-9$ .

Factors of $-12$	Sum of factors
$1,-12$	$1+(-12)=-11$
$1,12$	$-1+12=11$
$2,-6$	$2+(-6)=-4$
$-2,6$	$-2+6=4$
$3,-4$	$3+(-4)=-1$
$-3,4$	$-3+4=1$

Note there are no factor pairs that give us $-9$ as a sum. The trinomial is prime.

TRY IT 12.1

Factor: $x to the 2-7xy-10y to the 2$ .

Show answer

prime

TRY IT 12.2

Factor: $p to the 2+15pq+20q to the 2$ .

Show answer

prime

Key Concepts

Factor trinomials of the form $x to the 2+bx+c$
1. Write the factors as two binomials with first terms x: $(x)(x)$ .
2. Find two numbers m and n that
  Multiply to c, $m times n=c$
  Add to b, $m+n=b$
3. Use m and n as the last terms of the factors: $(x+m)(x+n)$ .
4. Check by multiplying the factors.

Practice Makes Perfect

Factor Trinomials of the Form $x to the 2+bx+c$

In the following exercises, factor each trinomial of the form $x to the 2+bx+c$ .

1. $x to the 2+4x+3$	2. $y to the 2+8y+7$
3. $m to the 2+12m+11$	4. $b to the 2+14b+13$
5. $a to the 2+9a+20$	6. $m to the 2+7m+12$
7. $p to the 2+11p+30$	8. $w to the 2+10x+21$
9. $n to the 2+19n+48$	10. $b to the 2+14b+48$
11. $a to the 2+25a+100$	12. $u to the 2+101u+100$
13. $x to the 2-8x+12$	14. $q to the 2-13q+36$
15. $y to the 2-18x+45$	16. $m to the 2-13m+30$
17. $x to the 2-8x+7$	18. $y to the 2-5y+6$
19. $p to the 2+5p-6$	20. $n to the 2+6n-7$
21. $y to the 2-6y-7$	22. $v to the 2-2v-3$
23. $x to the 2-x-12$	24. $r to the 2-2r-8$
25. $a to the 2-3a-28$	26. $b to the 2-13b-30$
27. $w to the 2-5w-36$	28. $t to the 2-3t-54$
29. $x to the 2+x+5$	30. $x to the 2-3x-9$
31. $8-6x+x to the 2$	32. $7x+x to the 2+6$
33. $x to the 2-12-11x$	34. $-11-10x+x to the 2$

Factor Trinomials of the Form $x to the 2+bxy+cy to the 2$

In the following exercises, factor each trinomial of the form $x to the 2+bxy+cy to the 2$ .

35. $p to the 2+3pq+2q to the 2$	36. $m to the 2+6mn+5n to the 2$
37. $r to the 2+15rs+36s to the 2$	38. $u to the 2+10uv+24v to the 2$
39. $m to the 2-12mn+20n to the 2$	40. $p to the 2-16pq+63q to the 2$
41. $x to the 2-2xy-80y to the 2$	42. $p to the 2-8pq-65q to the 2$
43. $m to the 2-64mn-65n to the 2$	44. $p to the 2-2pq-35q to the 2$
45. $a to the 2+5ab-24b to the 2$	46. $r to the 2+3rs-28s to the 2$
47. $x to the 2-3xy-14y to the 2$	48. $u to the 2-8uv-24v to the 2$
49. $m to the 2-5mn+30n to the 2$	50. $c to the 2-7cd+18d to the 2$

Mixed Practice

In the following exercises, factor each expression.

51. $u to the 2-12u+36$	52. $w to the 2+4w-32$
53. $x to the 2-14x-32$	54. $y to the 2+41y+40$
55. $r to the 2-20rs+64s to the 2$	56. $x to the 2-16xy+64y to the 2$
57. $k to the 2+34k+120$	58. $m to the 2+29m+120$
59. $y to the 2+10y+15$	60. $z to the 2-3z+28$
61. $m to the 2+mn-56n to the 2$	62. $q to the 2-29qr-96r to the 2$
63. $u to the 2-17uv+30v to the 2$	64. $m to the 2-31mn+30n to the 2$
65. $c to the 2-8cd+26d to the 2$	66. $r to the 2+11rs+36s to the 2$

Everyday Math

67. Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial $x to the 2+x-56$ describes how these numbers are related. Factor the trinomial.

68. Consecutive integers Deshawn is thinking of two consecutive integers whose product is 182. The trinomial $x to the 2+x-182$ describes how these numbers are related. Factor the trinomial.

Writing Exercises

69. Many trinomials of the form $x to the 2+bx+c$ factor into the product of two binomials $(x+m)(x+n)$ . Explain how you find the values of m and n.	70. How do you determine whether to use plus or minus signs in the binomial factors of a trinomial of the form $x to the 2+bx+c$ where $b$ and $c$ may be positive or negative numbers?
71. Will factored $x to the 2-x-20$ as $(x+5)(x-4)$ . Bill factored it as $(x+4)(x-5)$ . Phil factored it as $(x-5)(x-4)$ . Who is correct? Explain why the other two are wrong.	72. Look at (Figure), where we factored $y to the 2+17y+60$ . We made a table listing all pairs of factors of 60 and their sums. Do you find this kind of table helpful? Why or why not?

Answers

1. $(x+1)(x+3)$	3. $(m+1)(m+11)$
5. $(a+4)(a+5)$	7. $(p+5)(p+6)$
9. $(n+3)(n+16)$	11. $(a+5)(a+20)$
13. $(x-2)(x-6)$	15. $(y-3)(y-15)$
17. $(x-1)(x-7)$	19. $(p-1)(p+6)$
21. $(y+1)(y-7)$	23. $(x-4)(x+3)$
25. $(a-7)(a+4)$	27. $(w-9)(w+4)$
29. prime	31. $(x-4)(x-2)$
33. $(x-12)(x+1)$	35. $(p+q)(p+2q)$
37. $(r+3s)(r+12s)$	39. $(m-2n)(m-10n)$
41. $(x+8y)(x-10y)$	43. $(m+n)(m-65n)$
45. $(a+8b)(a-3b)$	47. prime
49. prime	51. $(u-6)(u-6)$
53. $(x+2)(x-16)$	55. $(r-4s)(r-16s)$
57. $(k+4)(k+30)$	59. prime
61. $(m+8n)(m-7n)$	63. $(u-15v)(u-2v)$
65. prime	67. $(x+8)(x-7)$
69. Answers may vary	71. Answers may vary

Attributions

This chapter has been adapted from “Factor Trinomials of the Form $x to the 2+bx+c$ ” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

8.6 Divide Polynomials

Learning Objectives

By the end of this section, you will be able to:

Divide a polynomial by a monomial

Divide a Polynomial by a Monomial

In the last chapter, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.

The sum,	$y over 5+2 over 5$ ,
simplifies to	$y+2 over 5$ .

Now we will do this in reverse to split a single fraction into separate fractions.

We’ll state the fraction addition property here just as you learned it and in reverse.

Fraction Addition

If $a,b$ , and $c$ are numbers where $c not equal to 0$ , then

$a over c+b over c=a+b over c$ and $a+b over c=a over c+b over c$

We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.

For example,	$y+2 over 5$
can be written	$y over 5+2 over 5$ .

We use this form of fraction addition to divide polynomials by monomials.

Division of a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

EXAMPLE 1

Find the quotient: $7y to the 2+21 over 7$ .

Solution

	$7y to the 2+21 over 7$
Divide each term of the numerator by the denominator.	$7y to the 2 over 7+21 over 7$
Simplify each fraction.	$y to the 2+3$

TRY IT 1.1

Find the quotient: $8z to the 2+24 over 4$ .

Show answer

$2z to the 2+6$

TRY IT 1.2

Find the quotient: $18z to the 2-27 over 9$ .

Show answer

$2z to the 2-3$

Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.

EXAMPLE 2

Find the quotient: $(18x to the 3-36x to the 2) divided by 6x$ .

Solution

	$(18x to the 3-36x to the 2) divided by 6x$
Rewrite as a fraction.	$18x to the 3-36x to the 2 over 6x$
Divide each term of the numerator by the denominator.	$18x to the 3 over 6x-36x to the 2 over 6x$
Simplify.	$3x to the 2-6x$

TRY IT 2.1

Find the quotient: $(27b to the 3-33b to the 2) divided by 3b$ .

Show answer

$9b to the 2-11b$

TRY IT 2.2

Find the quotient: $(25y to the 3-55y to the 2) divided by 5y$ .

Show answer

$5y to the 2-11y$

When we divide by a negative, we must be extra careful with the signs.

EXAMPLE 3

Find the quotient: $12d to the 2-16d over -4$ .

Solution

	$12d to the 2-16d over -4$
Divide each term of the numerator by the denominator.	$12d to the 2 over -4-16d over -4$
Simplify. Remember, subtracting a negative is like adding a positive!	$-3d to the 2+4d$

TRY IT 3.1

Find the quotient: $25y to the 2-15y over -5$ .

Show answer

$-5y to the 2+3y$

TRY IT 3.2

Find the quotient: $42b to the 2-18b over -6$ .

Show answer

$-7b to the 2+3b$

EXAMPLE 4

Find the quotient: $105y to the 5+75y to the 3 over 5y to the 2$ .

Solution

	$105y to the 5+75y to the 3 over 5y to the 2$
Separate the terms.	$105y to the 5 over 5y to the 2+75y to the 3 over 5y to the 2$
Simplify.	$21y to the 3+15y$

TRY IT 4.1

Find the quotient: $60d to the 7+24d to the 5 over 4d to the 3$ .

Show answer

$15d to the 4+6d to the 2$

TRY IT 4.2

Find the quotient: $216p to the 7-48p to the 5 over 6p to the 3$ .

Show answer

$36p to the 4-8p to the 2$

EXAMPLE 5

Find the quotient: $(15x to the 3y-35xy to the 2) divided by (-5xy)$ .

Solution

	$(15x to the 3y-35xy to the 2) divided by (-5xy)$
Rewrite as a fraction.	$15x to the 3y-35xy to the 2 over -5xy$
Separate the terms.	$15x to the 3y over -5xy-35xy to the 2 over -5xy$
Simplify.	$-3x to the 2+7y$

TRY IT 5.1

Find the quotient: $(32a to the 2b-16ab to the 2) divided by (-8ab)$ .

Show answer

$-4a+2b$

TRY IT 5.2

Find the quotient: $(-48a to the 8b to the 4-36a to the 6b to the 5) divided by (-6a to the 3b to the 3)$ .

Show answer

$8a to the 5b+6a to the 3b to the 2$

EXAMPLE 6

Find the quotient: $36x to the 3y to the 2+27x to the 2y to the 2-9x to the 2y to the 3 over 9x to the 2y$ .

Solution

	$36x to the 3y to the 2+27x to the 2y to the 2-9x to the 2y to the 3 over 9x to the 2y$
Separate the terms.	$36x to the 3y to the 2 over 9x to the 2y+27x to the 2y to the 2 over 9x to the 2y-9x to the 2y to the 3 over 9x to the 2y$
Simplify.	$4xy+3y-y to the 2$

TRY IT 6.1

Find the quotient: $40x to the 3y to the 2+24x to the 2y to the 2-16x to the 2y to the 3 over 8x to the 2y$ .

Show answer

$5xy+3y-2y to the 2$

TRY IT 6.2

Find the quotient: $35a to the 4b to the 2+14a to the 4b to the 3-42a to the 2b to the 4 over 7a to the 2b to the 2$ .

Show answer

$5a to the 2+2a to the 2b-6b to the 2$

EXAMPLE 7

Find the quotient: $10x to the 2+5x-20 over 5x$ .

Solution

	$10x to the 2+5x-20 over 5x$
Separate the terms.	$10x to the 2 over 5x+5x over 5x-20 over 5x$
Simplify.	$2x+1+4 over x$

TRY IT 7.1

Find the quotient: $18c to the 2+6c-9 over 6c$ .

Show answer

$3c+1-3 over 2c$

TRY IT 7.2

Find the quotient: $10d to the 2-5d-2 over 5d$ .

Show answer

$2d-1-2 over 5d$

Access these online resources for additional instruction and practice with dividing polynomials:

Key Concepts

Fraction Addition
- If $a,b$ , and $c$ are numbers where $c not equal to 0$ , then
  $a over c+b over c=a+b over c$ and $a+b over c=a over c+b over c$
Division of a Polynomial by a Monomial
- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Practice Makes Perfect

Dividing Polynomial by Monomial

In the following exercises, divide each polynomial by the monomial.

1. $30b+75 over 5$	2. $45y+36 over 9$
3. $42x to the 2-14x over 7$	4. $8d to the 2-4d over 2$
5. $(55w to the 2-10w) divided by 5w$	6. $(16y to the 2-20y) divided by 4y$
7. $(8x to the 3+6x to the 2) divided by 2x$	8. $(9n to the 4+6n to the 3) divided by 3n$
9. $20b to the 2-12b over -4$	10. $18y to the 2-12y over -6$
11. $51m to the 4+72m to the 3 over -3$	12. $35a to the 4+65a to the 2 over -5$
13. $412z to the 8-48z to the 5 over 4z to the 3$	14. $310y to the 4-200y to the 3 over 5y to the 2$
15. $51y to the 4+42y to the 2 over 3y to the 2$	16. $46x to the 3+38x to the 2 over 2x to the 2$
17. $(35x to the 4-21x) divided by (-7x)$	18. $(24p to the 2-33p) divided by (-3p)$
19. $(48y to the 4-24y to the 3) divided by (-8y to the 2)$	20. $(63m to the 4-42m to the 3) divided by (-7m to the 2)$
21. $(45x to the 3y to the 4+60xy to the 2) divided by (5xy)$	22. $(63a to the 2b to the 3+72ab to the 4) divided by (9ab)$
23. $49c to the 2d to the 2-70c to the 3d to the 3-35c to the 2d to the 4 over 7cd to the 2$	24. $52p to the 5q to the 4+36p to the 4q to the 3-64p to the 3q to the 2 over 4p to the 2q$
25. $72r to the 5s to the 2+132r to the 4s to the 3-96r to the 3s to the 5 over 12r to the 2s to the 2$	26. $66x to the 3y to the 2-110x to the 2y to the 3-44x to the 4y to the 3 over 11x to the 2y to the 2$
27. $12q to the 2+3q-1 over 3q$	28. $4w to the 2+2w-5 over 2w$
29. $20y to the 2+12y-1 over -4y$	30. $10x to the 2+5x-4 over -5x$
31. $63a to the 3-108a to the 2+99a over 9a to the 2$	32. $36p to the 3+18p to the 2-12p over 6p to the 2$

Everyday Math

33. Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression $n to the 2-n over 2$ , where $n$ represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

34. Average cost Pictures Plus produces digital albums. The company’s average cost (in dollars) to make $x$ albums is given by the expression $7x+500 over x$ .

Find the quotient by dividing the numerator by the denominator.
What will the average cost (in dollars) be to produce 20 albums?

Writing Exercises

35. Divide $10x to the 2+x-12 over 2x$ and explain with words how you get each term of the quotient.

36. James divides $48y+6$ by 6 this way: $mathematical expression$ . What is wrong with his reasoning?

Answers

1. $6b+15$	3. $6x to the 2-2x$	5. $11w-2$
7. $4x to the 2+3x$	9. $-5b to the 2+3b$	11. $-17m to the 4-24m to the 3$
13. $103z to the 5-12z to the 2$	15. $17y to the 2+14$	17. $-5x to the 3+3$
19. $-6y to the 2+3y$	21. $9x to the 2y to the 3+12y$	23. $7c-10c to the 2d-5cd to the 2$
25. $6r to the 3+11r to the 2s-8rs to the 3$	27. $4q+1-1 over 3q$	29. $-5y-3+1 over 4y$
31. $7a-12+11 over a$	33. 45	35. Answers will vary.

Attributions

This chapter has been adapted from “Divide Polynomials” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

8.7 Chapter Review

Review Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

a) $a to the 2-b to the 2$
b) $24d to the 3$
c) $x to the 2+8x-10$
d) $m to the 2n to the 2-2mn+6$
e) $7y to the 3+y to the 2-2y-4$

a) $11c to the 4-23c to the 2+1$
b) $9p to the 3+6p to the 2-p-5$
c) $3 over 7x+5 over 14$
d) 10
e) $2y-12$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

a) $5p to the 3-8p to the 2+10p-4$

b) $-20q to the 4$

c) $x to the 2+6x+12$

d) $23r to the 2s to the 2-4rs+5$

e) 100

a) $3x to the 2+9x+10$

b) $14a to the 2bc$

c) $6y+1$

d) $n to the 3-4n to the 2+2n-8$

e) $-19$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

5. $-14k+19k$	6. $mathematical expression$
7. $-9c-18c$	8. $12q-(-6q)$
9. $3m to the 2+7n to the 2-3m to the 2$	10. $12x-4y-9x$
11. $13a+b$	12. $6x to the 2y-4x+8xy to the 2$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

13. $(9p to the 2-5p+3)+(4p to the 2-4)$	14. $(5x to the 2+12x+1)+(6x to the 2-8x+3)$
15. $(7y to the 2-8y)-(y-4)$	16. $(10m to the 2-8m-1)-(5m to the 2+m-2)$
17. Find the sum of $mathematical expression$	18. Subtract $mathematical expression$

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

19. Evaluate $10-12x$ when:

a) $x=3$

b) $x=0$

c) $x=-1$

20. Evaluate $3y to the 2-y+1$ when:

a) $y=5$

b) $y=-1$

c) $y=0$

21. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $-4p to the 2+460p$ . Find the revenue received when $p=75$ dollars.

22. Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial $-16t to the 2+200$ gives the height of a stone $t$ seconds after it is dropped from the cliff. Find the height after $t=3$ seconds.

Multiply Monomials

In the following exercises, multiply the monomials.

23. $(-9n to the 7)(-16n)$	24. $(-15x to the 2)(6x to the 4)$
25. $(5 over 9ab to the 2)(27ab to the 3)$	26. $(7p to the 5q to the 3)(8pq to the 9)$

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

27. $-4(y+13)$	28. $7(a+9)$
29. $p(p+3)$	30. $-5(r-2)$
31. $-6u(2u+7)$	32. $-m(m+15)$
33. $3q to the 2(q to the 2-7q+6)$ 3	34. $9(b to the 2+6b+8)$
35. $(b-4) times 11$	36. $(5z-1)z$

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: a) the Distributive Property, b) the FOIL method, c) the Vertical Method.

37. $(6y-7)(2y-5)$

38. $(x-4)(x+10)$

In the following exercises, multiply the binomials. Use any method.

39. $(y-4)(y-8)$	40. $(x+3)(x+9)$
41. $(q+16)(q-3)$	42. $(p-7)(p+4)$
43. $(u to the 2+6)(u to the 2-5)$	44. $(5m-8)(12m+1)$
45. $(8mn+3)(2mn-1)$	46. $(9x-y)(6x-5)$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using a) the Distributive Property, b) the Vertical Method.

47. $(3x-4)(6x to the 2+x-10)$

48. $(n+1)(n to the 2+5n-2)$

In the following exercises, multiply. Use either method.

49. $(7m+1)(m to the 2-10m-3)$

50. $(y-2)(y to the 2-8y+9)$

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

51. $(q-15) to the 2$	52. $(c+11) to the 2$
53. $(8u+1) to the 2$	54. $(x+1 over 3) to the 2$
55. $(4a-3b) to the 2$	56. $(3n to the 3-2) to the 2$

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

57. $(y+2 over 5)(y-2 over 5)$	58. $(s-7)(s+7)$
59. $(6-r)(6+r)$	60. $(12c+13)(12c-13)$
61. $(5p to the 4-4q to the 3)(5p to the 4+4q to the 3)$	62. $(u+3 over 4v)(u-3 over 4v)$

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

63. $(6a+11)(6a-11)$	64. $(3m+10) to the 2$
65. $(c to the 4+9d) to the 2$	66. $(5x+y)(x-5y)$
67. $(a to the 2+4b)(4a-b to the 2)$	68. $(p to the 5+q to the 5)(p to the 5-q to the 5)$

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

69. $(35x to the 2-75x) divided by 5x$	70. $42z to the 2-18z over 6$
71. $550p to the 6-300p to the 4 over 10p to the 3$	72. $81n to the 4+105n to the 2 over -3$
73. $96a to the 5b to the 2-48a to the 4b to the 3-56a to the 2b to the 4 over 8ab to the 2$	74. $(63xy to the 3+56x to the 2y to the 4) divided by (7xy)$
75. $105y to the 5+50y to the 3-5y over 5y to the 3$	76. $57m to the 2-12m+1 over -3m$

Review Exercise Answers

1. a) binomial b) monomial c) trinomial d) trinomial e) other polynomial	3. a) 3 b) 4 c) 2 d) 4 e) 0	5. $5k$
7. $-27c$	9. $7n to the 2$	11. $13a+b$
13. $13p to the 2-5p-1$	15. $7y to the 2-9y+4$	17. $5a to the 3+a to the 2+6a+2$
19. a) $-26$ b) 10 c) 22	21. 12,000	23. $144n to the 8$
25. $15a to the 2b to the 5$	27. $-4y-52$	29. $p to the 2+3p$
31. $-12u to the 2-42u$	33. $3q to the 4-21q to the 3+18q to the 2$	35. $11b-44$
37. a) $12y to the 2-44y+35$ b) $12y to the 2-44y+35$ c) $12y to the 2-44y+35$	39. $y to the 2-12y+32$	41. $q to the 2+13q-48$
43. $u to the 4+u to the 2-30$	45. $16m to the 2n to the 2-2mn-3$	47. a) $18x to the 3-21x to the 2-34x+40$ b) $18x to the 3-21x to the 2-34x+40$
49. $7m to the 3-69m to the 2-31m-3$	51. $q to the 2-30q+225$	53. $64u to the 2+16u+1$
55. $16a to the 2-24ab+9b to the 2$	57. $y to the 2-4 over 25$	59. $36-r to the 2$
61. $25p to the 8-16q to the 6$	63. $36a to the 2-121$	65. $c to the 8+18c to the 4d+81d to the 2$
67. $4a to the 3+3a to the 2b-4b to the 3$	69. $7x-15$	71. $55p to the 3-30p$
73. $12a to the 4-6a to the 3b-7ab to the 2$	75. $21y to the 2+10-1 over y to the 2$

Chapter Practice Test

In the following exercises, simplify each expression. 1. $(12a to the 2-7a+4)+(3a to the 2+8a-10)$	2. For the polynomial $10x to the 4+9y to the 2-1$ a) Is it a monomial, binomial, or trinomial? b) What is its degree?
3. $(9p to the 2-5p+1)-(2p to the 2-6)$	4. $(-9r to the 4s to the 5)(4rs to the 7)$
5. $(v-9)(9v-5)$	6. $(m+6)(m+12)$
7. $(n-6)(n to the 2-5n+4)$	8. $(4c-11)(3c-8)$
9. $(7p-5)(7p+5)$	10. $(2x-15y)(5x+7y)$
11. $(9v-2) to the 2$	12. $12x to the 3+42x to the 2-6x over 2x$
13. $64x to the 3-x over 4x$	14. $70xy to the 4+95x to the 3y over 5xy$
15. $y to the 2-5y-18 over y$	16. A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $-16t to the 2+1000$ gives the height of the package $t$ seconds after it was dropped. Find the height when $t=6$ seconds.

Practice Test Answers

1. $15a to the 2+a-6$	2. a) Trinomial, b) 4	3. $7p to the 2-5p+7$
4. $-36r to the 5s to the 12$	5. $9v to the 2-86v+45$	6. $m to the 2+18m+72$
7. $n to the 3-11n to the 2+34n-24$	8. $12c to the 2-65c+88$	9. $49p to the 2-25$
10. $10x to the 2-61xy-105y to the 2$	11. $81v to the 2-36v+4$	12. $6x to the 2+21x-3$
13. $16x to the 2-1 over 4$	14. $14 y to the 3+19x to the 2$	15. $y -5-$ $18 over y$
16. 424 feet

	$(5x-6)(6x+5)$
Use FOIL.	$30x to the 2+25x-36x-30$
Simplify.	$30x to the 2-11x-30$