8.2 Multiply Polynomials

Izabela Mazur

CHAPTER 8 Polynomials

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

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.

$\begin{array}{c}\hfill \left(x-2\right)\left(x-y\right)\hfill \\ \hfill {x}^{2}-xy-2x+2y\hfill \end{array}$

Where did the first term, ${x}^{2}$ , 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.

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

$\begin{matrix}{x}^{2}- xy- 2x+ 2y \\ F \quad \enspace O \quad \enspace I \quad \enspace L \end{matrix}$

Let’s look at $\left(x+3\right)\left(x+7\right)$ .

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: $\left(x+5\right)\left(x+9\right)$ .

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: $\left(x+6\right)\left(x+8\right)$ .

Show answer

${x}^{2}+14x+48$

TRY IT 11.2

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

Show answer

${y}^{2}+20y+51$

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: $\left(y-7\right)\left(y+4\right)$ .

Solution

TRY IT 12.1

Multiply: $\left(x-7\right)\left(x+5\right)$ .

Show answer

${x}^{2}-2x-35$

TRY IT 12.2

Multiply: $\left(b-3\right)\left(b+6\right)$ .

Show answer

${b}^{2}+3b-18$

EXAMPLE 13

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

Solution

TRY IT 13.1

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

Show answer

$15{x}^{2}+29x-14$

TRY IT 13.2

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

Show answer

$16{y}^{2}-20y-50$

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: $\left(3x-y\right)\left(2x-5\right)$ .

Solution



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

TRY IT 14.1

Multiply: $\left(10c-d\right)\left(c-6\right)$ .

Show answer

$10{c}^{2}-60c-cd+6d$

TRY IT 14.2

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

Show answer

$14{x}^{2}-35x-2xy+10y$

Be careful of the exponents in the next example.

EXAMPLE 15

Multiply: $\left({n}^{2}+4\right)\left(n-1\right)$ .

Solution



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

TRY IT 15.1

Multiply: $\left({x}^{2}+6\right)\left(x-8\right)$ .

Show answer

${x}^{3}-8{x}^{2}+6x-48$

TRY IT 15.2

Multiply: $\left({y}^{2}+7\right)\left(y-9\right)$ .

Show answer

${y}^{3}-9{y}^{2}+7y-63$

EXAMPLE 16

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

Solution


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

TRY IT 16.1

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

Show answer

$8{a}^{2}{b}^{2}+12ab-20$

TRY IT 16.2

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

Show answer

$8{x}^{2}{y}^{2}+2xy-15$

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

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

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

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

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

Multiply a Trinomial by a Binomial

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

95. $\left(x+5\right)\left({x}^{2}+4x+3\right)$	96. $\left(u+4\right)\left({u}^{2}+3u+2\right)$
97. $\left(y+8\right)\left(4{y}^{2}+y-7\right)$	98. $\left(a+10\right)\left(3{a}^{2}+a-5\right)$

In the following exercises, multiply. Use either method.

99. $\left(w-7\right)\left({w}^{2}-9w+10\right)$	100. $\left(p-4\right)\left({p}^{2}-6p+9\right)$
101. $\left(3q+1\right)\left({q}^{2}-4q-5\right)$	102. $\left(6r+1\right)\left({r}^{2}-7r-9\right)$

Mixed Practice

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

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 $\left(10+3\right)\left(10+5\right)$ by the FOIL method.
Multiply $13\cdot 15$ 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 $\left(20-2\right)\left(20-3\right)$ by the FOIL method.
Multiply $18\cdot 17$ 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:

$\begin{array}{c}\left(x+2\right)\left(x-2\right)\hfill \\ \left(y+7\right)\left(y-7\right)\hfill \\ \left(w+5\right)\left(w-5\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

127. Multiply the following:

$\begin{array}{c}\left(m-3\right)\left(m+3\right)\hfill \\ \left(n-10\right)\left(n+10\right)\hfill \\ \left(p-8\right)\left(p+8\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

128. Multiply the following:

$\begin{array}{c}\left(p+3\right)\left(p+3\right)\hfill \\ \left(q+6\right)\left(q+6\right)\hfill \\ \left(r+1\right)\left(r+1\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

129. Multiply the following:

$\begin{array}{c}\left(x-4\right)\left(x-4\right)\hfill \\ \left(y-1\right)\left(y-1\right)\hfill \\ \left(z-7\right)\left(z-7\right)\hfill \end{array}$

Explain the pattern that you see in your answers.

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


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

Multiply (2b² − 5b + 8) by 3.

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


Distribute.
Simplify.


Distribute.
Simplify.


Distribute.
Simplify.


Distribute.
Simplify.

Multiply a Polynomial by a Monomial

Multiply a Binomial by a Binomial

Multiply a Binomial by a Binomial Using the Distributive Property

Multiply a Binomial by a Binomial Using the FOIL Method

Multiply a Binomial by a Binomial Using the Vertical Method

Multiply a Trinomial by a Binomial

Key Concepts

Practice Makes Perfect

Multiply a Polynomial by a Monomial

Multiply a Binomial by a Binomial

Multiply a Trinomial by a Binomial

Mixed Practice

Everyday Math

Writing Exercises

Answers:

Attributions

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