# 6.3 Multiply polynomials

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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

Before you get started, take this readiness quiz.

1. Distribute: $2\left(x+3\right).$
If you missed this problem, review [link] .
2. Combine like terms: ${x}^{2}+9x+7x+63.$
If you missed this problem, review [link] .

## Multiply a polynomial by a monomial

We have used the Distributive Property to simplify expressions like $2\left(x-3\right)$ . You multiplied both terms in the parentheses, $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3$ , 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:

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

## Solution

 Distribute. Simplify.

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

$5x+35$

Multiply: $3\left(y+13\right).$

$3y+39$

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

## Solution

 Distribute. Simplify.

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

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

Multiply: $d\left(d-11\right).$

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

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

## Solution

 Distribute. Simplify.

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

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

Multiply: $2p\left(6p+r\right).$

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

Multiply: $-2y\left(4{y}^{2}+3y-5\right).$

## Solution

 Distribute. Simplify.

Multiply: $-3y\left(5{y}^{2}+8y-7\right).$

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

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

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

Multiply: $2{x}^{3}\left({x}^{2}-8x+1\right).$

## Solution

 Distribute. Simplify.

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

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

Multiply: $-6{a}^{3}\left(3{a}^{2}-2a+6\right).$

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

Multiply: $\left(x+3\right)p.$

## Solution

 The monomial is the second factor. Distribute. Simplify.

Multiply: $\left(x+8\right)p.$

$xp+8p$

Multiply: $\left(a+4\right)p.$

$ap+4p$

## 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 [link] , where we multiplied a binomial by a monomial    .

 We distributed the p to get: What if we have ( x + 7) instead of p ? Distribute ( x + 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.

Multiply: $\left(y+5\right)\left(y+8\right).$

## Solution

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

Multiply: $\left(x+8\right)\left(x+9\right).$

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

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

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

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

## Solution

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

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

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

Multiply: $\left(a+10\right)\left(a+7\right).$

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

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

## Solution

 Distribute. Distribute again. Combine like terms.

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

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

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

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

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

## Solution

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

Multiply: $\left(a+7\right)\left(a-b\right).$

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

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

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

## 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 [link] , 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 ‘ F irst, O uter, I nner, L ast’. The word FOIL is easy to remember and ensures we find all four products.

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