# 5.3 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

## Try it

Before you get started, take this readiness quiz:

1) Distribute: $2(x+3)$
2) Combine like terms: ${x}^{2}+9x+7x+63$

## 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 5.3.1

Multiply: $4(x+3)$

Solution

Step 1: Distribute.

\begin{align*} &4\;\times\;x\;+\;4\;\times\;3\\ \text{Simplify.}\;\;&4x\;+\;12 \end{align*}

### Try It

3) Multiply: $5(x+7)$

Solution

$5x+35$

4) Multiply: $3(y+13)$

Solution

$3y+39$

### Example 5.3.2

Multiply: $y(y-2)$

Solution

Step 1: Distribute.

\begin{align*} &y\;\times\;y\;-\;y\;\times\;2\\ \text{Simplify.}\;\;&y^2\;-\;2y \end{align*}

### Try It

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

Solution

${x}^{2}-7x$

6) Multiply: $d(d-11)$

Solution

${d}^{2}-11d$

### Example 5.3.3

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

Solution

Step 1: Distribute.

\begin{align*} &7x\;\times\;2x\;+\;7x\;\times\;y\\ \text{Simplify.}\;\;&14x^2\;+\;7xy \end{align*}

### Try It

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

Solution

$5{x}^{2}+20xy$

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

Solution

$12{p}^{2}+2pr$

### Example 5.3.4

Multiply: $-2y(4{y}^{2}+3y-5)$

Solution

Step 1: Distribute.

\begin{align*} &-2y\;\times\;4y^2\;+\;(-2y)\;\times\;3y\;-\;(-2y)\;\times\;5\\ \text{Simplify.}\;\;&-8y^3\;-\;6y^2\;+\;10y \end{align*}

### Try It

9) Multiply: $-3y(5{y}^{2}+8y-7)$

Solution

$-15{y}^{3}-24{y}^{2}+21y$

10) Multiply: $4{x}^{2}(2{x}^{2}-3x+5)$

Solution

$8{x}^{4}-24{x}^{3}+20{x}^{2}$

### Example 5.3.5

Multiply:  $2{x}^{3}({x}^{2}-8x+1)$

Solution

Step 1: Distribute.

\begin{align*} &2x^3\;\times\;x^2\;+\;(2x^3)\;\times\;(-8x)\;+\;(2x^3)\;\times\;1\\ \text{Simplify.}\;\;&2x^5\;-\;16x^4\;+\;2x^3 \end{align*}

### Try It

11) Multiply: $4x(3{x}^{2}-5x+3)$

Solution

$12{x}^{3}-20{x}^{2}+12x$

12) Multiply: $-6{a}^{3}(3{a}^{2}-2a+6)$

Solution

$-18{a}^{5}+12{a}^{4}-36{a}^{3}$

### Example 5.3.6

Multiply: $(x+3)p$.

Solution

Step 1: The monomial is the second factor.

Step 2: Distribute.

\begin{align*} &x\;\times\;p\;+\;3\;\times\;p\\ \text{Simplify.}\;\;&xp\;+\;3p \end{align*}

### Try It

13) Multiply: $(x+8)p$

Solution

$xp+8p$

14) Multiply: $(a+4)p$

Solution

$ap+4p$

## 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 below table, where we multiplied a binomial by a monomial.

 We distributed the $p$ to get: $x{\color{red}{p}}+3{\color{red}{p}}$ What if we have $(x + 7)$ instead of $p$? Distribute $(x + 7)$. Distribute again. $x^2+7x+3x+21$ Combine like terms. $x^2+10x+21$

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 5.3.7

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

Solution

Step 1: Distribute $(y + 8)$.

$y{\color{red}{(y\;+\;8)}}\;+\;5{\color{red}{(y\;+\;8)}}$

Step 2: Distribute again

$y^2\;+\;8y\;+\;5y\;+\;40$

Step 3: Combine like terms.

$y^2\;+\;13y\;+\;40$

### Try It

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

Solution

${x}^{2}+17x+72$

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

Solution

$20{x}^{2}+51x+27$

### Example 5.3.8

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

Solution

Step 1: Distribute $(3y + 4)$.

${2y}\;{\color{red}{(3y\;+\;4)}}\;+\;5\;{\color{red}{(3y\;+\;4)}}$

Step 2: Distribute again

$6y^2\;+\;8y\;+\;15y\;+\;20$

Step 3: Combine like terms.

$6y^2\;+\;23y\;+\;20$

### Try It

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

Solution

$12{b}^{2}+38b+30$

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

Solution

${a}^{2}+17a+70$

### Example 5.3.9

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

Solution

Step 1: Distribute.

$4y\;{\color{red}{(2y\;-\;5)}}\;+\;3\;{\color{red}{(2y\;-\;5)}}$

Step 2: Distribute again.

$8y^2\;-\;20y\;+\;6y\;-\;15$

Step 3: Combine like terms.

$8y^2\;-\;14y\;-\;15$

### Try It

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

Solution

$30{y}^{2}-3y-6$

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

Solution

$15{c}^{2}+14c-8$

### Example 5.3.10

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

Solution

Step 1: Distribute.

${x}{\color{red}{(x\;-\;y)\;-}}\;2{\color{red}{(x\;-\;y)}}$

Step 2: Distribute again.

$x^2\;-\;xy\;-\;2x\;+\;2y$

Step 3: There are no like terms to combine.

### Try It

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

Solution

${a}^{2}-ab+7a-7b$

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

Solution

${x}^{2}-xy+5x-5y$

## 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 Example 5.3.10  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.

$(x-2)(x-y)$
$x^2 - xy -2x +2y$

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.

$(x-2)(x-y)$
$x^2-xy-2x+2y$
F.   O.   I.    L

Let’s look at $(x+3)(x+7)$.

Distributive Property FOIL
$(x+3)(x+7)$
$x(x+7)+3(x+7)$
$\begin{array}{llllllll} &x^2&+&7x&+&3x&+&21&\\ &F&&O&&I&&L \end{array}$ $\begin{array}{llllllll} &x^2&+&7x&+&3x&+&21&\\ &F&&O&&I&&L \end{array}$
$x^2+10x+21$ $x^2+10x+21$

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 5.3.11

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

Solution

### Try It

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

Solution

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

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

Solution

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

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

## Multiply two binomials using the FOIL method

### HOW TO

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 5.3.12

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

Solution

### Try It

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

Solution

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

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

Solution

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

### Example 5.3.13

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

Solution

### Try It

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

Solution

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

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

Solution

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

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 5.3.14

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

Solution

Step 1: Multiply the First.

$\underset{F}{\color{red}{6x^2}}\;+\;\underset{O}{\_\_}\;+\;\underset{I}{\_\_}\;+\;\underset{L}{\;\_\_}$

Step 2: Multiply the Outer.

$\underset{F}{\color{red}{6x^2}}\;{\color{red}{-}}\;\underset{O}{\color{red}{15x}}\;+\;\underset{I}{\_\_}\;+\;\underset{L}{\;\_\_}$

Step 3: Multiply the Inner.

$\underset{F}{6x^2}\;-\;\underset{O}{15x}\;{\color{red}{-}}\;\underset{I}{\color{red}{2xy}}\;+\;\underset{L}{\;\_\_}$

Step 4: Multiply the Last.

$\underset{F}{6x^2}\;-\;\underset{O}{15x}\;-\;\underset{I}{2xy}\;{\color{red}{+}}\;\underset{L}{\color{red}{5y}}$

Step 5: Combine like terms—there are none.

$6x^2\;-\;15x\;-\;2xy\;+\;5y$

### Try It

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

Solution

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

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

Solution

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

Be careful of the exponents in the next example.

### Example 5.3.15

Multiply: $({n}^{2}+4)(n-1)$

Solution

Step 1: Multiply the First.

$\underset{F}{\color{red}{n^3}}\;+\;\underset{O}{\_\_}\;+\;\underset{I}{\_\_}\;+\;\underset{L}{\;\_\_}$

Step 2: Multiply the Outer.

$\underset{F}{n^3}\;{\color{red}{-}}\;\underset{O}{\color{red}{n^2}}\;+\;\underset{I}{\_\_}\;+\;\underset{L}{\;\_\_}$

Step 3: Multiply the Inner.

$\underset{F}{n^3}\;-\;\underset{O}{n^2}\;{\color{red}{+}}\;\underset{I}{\color{red}{4n}}\;+\;\underset{L}{\;\_\_}$

Step 4: Multiply the Last.

$\underset{F}{n^3}\;-\;\underset{O}{n^2}\;+\;\underset{I}{4n}\;{\color{red}{-}}\;\underset{L}{\color{red}{4}}$

Step 5: Combine like terms—there are none.

$n^3\;-\;n^2\;+\;4n\;-\;4$

### Try It

31) Multiply: $({x}^{2}+6)(x-8)$

Solution

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

32) Multiply: $({y}^{2}+7)(y-9)$

Solution

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

### Example 5.3.16

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

Solution

Step 1: Multiply the First.

$\underset F{\color{red}{18p^2q^2}}\;+\;\underset O{\_\_}\;+\;\underset I{\_\_}\;+\;\underset L{\_\_}$

Step 2: Multiply the Outer.

$\underset F{18p^2q^2}\;{\color{red}{-}}\;\underset{O}{\color{red}{33pq}}\;+\;\underset I{\_\_}\;+\;\underset L{\_\_}$

Step 3: Multiply the Inner.

$\underset F{18p^2q^2}\;-\;\underset O{33pq}\;{\color{red}{+}}\;\underset{I}{\color{red}{30pq}}\;+\;\underset L{\_\_}$

Step 4: Multiply the Last.

$\underset F{18p^2q^2}\;-\;\underset O{33pq}\;+\;\underset I{30pq}\;{\color{red}{-}}\;{\color{red}{\underbrace{55}_{\color{black}{L}}}}$

Step 5: Combine like terms—there are none.

$18p^2q^2\;-\;3pq\;-\;55\;$

### Try It

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

Solution

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

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

Solution

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

## 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.

\begin{alignat*}{3} &\;\;\;23\\ &\underline{\times46}\;\\ &\;\;138\;&&\;\text{partial product}\;\;\;\;\;&&\text{Start by multiplying 23 by 6 to get 138.}\\ &\underline{\;\;92\;\;}&&\;\text{partial product}\;\;\;\;\;&&\text{Next, multiply 23 by 4, lining up the partial product in the correct columns.}\\ &1058\;&&\;\text{product}\;\;\;\;\;&&\text{Last you add the partial products.} \end{alignat*}

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

### Example 5.3.17

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

Solution

It does not matter which binomial goes on the top.

Step 1: Multiply $3y -1$ by $-6$

Step 2: Multiply $3y-1$ by $2y$

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

 \begin{align*}(3y-1)(2y-6)\\{\color{blue}{\underline{6y2^2-2y}}}{\color{red}{\underline{\;-18y+6}}}\\6y^2-20y+6\end{align*} \begin{align*}3y-1\\\underline{\times2y-6}\\{\color{red}{-18y+6}}\\{\color{blue}{6y^2-2y}}\;\;\;\;\;\;\;\\\overline{6y^2-20x+6}\end{align*}

### Try It

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

Solution

$15{m}^{2}-51m+42$

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

Solution

$42{b}^{2}-53b+15$

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

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 5.3.18

Multiply using the Distributive Property: $(b+3)(2{b}^{2}-5b+8)$

Solution

Step 1: Distribute.

Step 2: Multiply.

$2b^3\;-\;5b^2\;+\;8b\;+\;6b^2\;-\;15b\;+\;24$

Step 3: Combine like terms.

$2b^3\;+\;b^2\;-\;7b\;+\;24$

### Try It

37) Multiply using the Distributive Property: $(y-3)({y}^{2}-5y+2)$

Solution

${y}^{3}-8{y}^{2}+17y-6$

38) Multiply using the Distributive Property: $(x+4)(2{x}^{2}-3x+5)$

Solution

$2{x}^{3}+5{x}^{2}-7x+20$

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

### Example 5.3.19

Multiply using the Vertical Method: $(b+3)(2{b}^{2}-5b+8)$

Solution

Step 1: Multiply $(2b^2-5b+8)$ by $3$.

\begin{align*} 2b^2-5b+8&\\ \times\;\;\;\;\;\;\;b+3&\\ \overline{6b^2-15b+24}& \end{align*}

Step 2: Multiply $(2b^2-5b+8)$ by $b$.

$2b^3\;-\;5b^2\;+\;8b$

$2b^3\;+\;b^2\;-\;7b\;+\;24$

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

### Try It

39) Multiply using the Vertical Method: $(y-3)({y}^{2}-5y+2)$

Solution

${y}^{3}-8{y}^{2}+17y-6$

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

Solution

$2x^3+5x^2-7x+20$

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

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 (Example 5.3.7)
• FOIL Method (Example 5.3.11)
• Vertical Method (Example 5.3.17)

• Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:
• Distributive Property (Example 5.3.18)
• Vertical Method (Example 5.3.19)

### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?