# 10.3 Multiply polynomials  (Page 3/3)

 Page 3 / 3 You start by multiplying $23$ by $6$ to get $138.$

Then you multiply $23$ by $4,$ lining up the partial product in the correct columns.

Last, you add the partial products.

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

Multiply using the vertical method: $\left(5x-1\right)\left(2x-7\right).$

## Solution

It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied $23\left(46\right).$ Multiply $2x-7$ by $-1$ . Multiply $2x-7$ by $5x$ . Add like terms. Notice the partial products are the same as the terms in the FOIL method. Multiply using the vertical method: $\left(4m-9\right)\left(3m-7\right).$

12 m 2 − 55 m + 63

Multiply using the vertical method: $\left(6n-5\right)\left(7n-2\right).$

42 n 2 − 47 n + 10

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

## Multiplying two binomials

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, the FOIL method 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.

Multiply using the Distributive Property: $\left(x+3\right)\left(2{x}^{2}-5x+8\right).$

## Solution Distribute. Multiply. $2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24$ Combine like terms. $2{x}^{3}+{x}^{2}-7x+24$

Multiply using the Distributive Property: $\left(y-1\right)\left({y}^{2}-7y+2\right).$

y 3 − 8 y 2 + 9 y − 2

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

3 x 3 + 2 x 2 − 3 x + 10

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

Multiply using the Vertical Method: $\left(x+3\right)\left(2{x}^{2}-5x+8\right).$

## Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way. Multiply $\left(2{x}^{2}-5x+8\right)$ by 3. Multiply $\left(2{x}^{2}-5x+8\right)$ by $x$ . Add like terms. Multiply using the Vertical Method: $\left(y-1\right)\left({y}^{2}-7y+2\right).$

y 3 − 8 y 2 + 9 y − 2

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

3 x 3 + 2 x 2 − 3 x + 10

## Key concepts

• Use the FOIL method for multiplying two binomials.  Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms. Step 5. Combine like terms, when possible.
• Multiplying Two Binomials: To multiply binomials, use the:
• Distributive Property
• FOIL Method
• Vertical Method
• Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
• Distributive Property
• Vertical Method

## Practice makes perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$4\left(x+10\right)$

4 x + 40

$6\left(y+8\right)$

$15\left(r-24\right)$

15 r − 360

$12\left(v-30\right)$

$-3\left(m+11\right)$

−3 m − 33

$-4\left(p+15\right)$

$-8\left(z-5\right)$

−8 z + 40

$-3\left(x-9\right)$

$u\left(u+5\right)$

u 2 + 5 u

$q\left(q+7\right)$

$n\left({n}^{2}-3n\right)$

n 3 − 3 n 2

$s\left({s}^{2}-6s\right)$

$12x\left(x-10\right)$

12 x 2 − 120 x

$9m\left(m-11\right)$

$-9a\left(3a+5\right)$

−27 a 2 − 45 a

$-4p\left(2p+7\right)$

$6x\left(4x+y\right)$

24 x 2 + 6 xy

$5a\left(9a+b\right)$

$5p\left(11p-5q\right)$

55 p 2 − 25 pq

$12u\left(3u-4v\right)$

$3\left({v}^{2}+10v+25\right)$

3 v 2 + 30 v + 75

$6\left({x}^{2}+8x+16\right)$

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

8 n 3 − 8 n 2 + 2 n

$3r\left(2{r}^{2}-6r+2\right)$

$-8y\left({y}^{2}+2y-15\right)$

−8 y 3 − 16 y 2 + 120 y

$-5m\left({m}^{2}+3m-18\right)$

$5{q}^{3}\left({q}^{2}-2q+6\right)$

5 q 5 − 10 q 4 + 30 q 3

$9{r}^{3}\left({r}^{2}-3r+5\right)$

$-4{z}^{2}\left(3{z}^{2}+12z-1\right)$

−12 z 4 − 48 z 3 + 4 z 2

$-3{x}^{2}\left(7{x}^{2}+10x-1\right)$

$\left(2y-9\right)y$

2 y 2 − 9 y

$\left(8b-1\right)b$

$\left(w-6\right)\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}8$

8 w − 48

$\left(k-4\right)\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}5$

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using different methods.

$\left(x+4\right)\left(x+6\right)$

x 2 + 10 x + 24

$\left(u+8\right)\left(u+2\right)$

$\left(n+12\right)\left(n-3\right)$

n 2 + 9 n − 36

$\left(y+3\right)\left(y-9\right)$

$\left(y+8\right)\left(y+3\right)$

y 2 + 11 y + 24

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

$\left(a+6\right)\left(a+16\right)$

a 2 + 22 a + 96

$\left(q+8\right)\left(q+12\right)$

$\left(u-5\right)\left(u-9\right)$

u 2 − 14 u + 45

$\left(r-6\right)\left(r-2\right)$

$\left(z-10\right)\left(z-22\right)$

z 2 − 32 z + 220

$\left(b-5\right)\left(b-24\right)$

$\left(x-4\right)\left(x+7\right)$

x 2 + 3 x − 28

$\left(s-3\right)\left(s+8\right)$

$\left(v+12\right)\left(v-5\right)$

v 2 + 7 v − 60

$\left(d+15\right)\left(d-4\right)$

$\left(6n+5\right)\left(n+1\right)$

6 n 2 + 11 n + 5

$\left(7y+1\right)\left(y+3\right)$

$\left(2m-9\right)\left(10m+1\right)$

20 m 2 − 88 m − 9

$\left(5r-4\right)\left(12r+1\right)$

$\left(4c-1\right)\left(4c+1\right)$

16 c 2 − 1

$\left(8n-1\right)\left(8n+1\right)$

$\left(3u-8\right)\left(5u-14\right)$

15 u 2 − 82 u + 112

$\left(2q-5\right)\left(7q-11\right)$

$\left(a+b\right)\left(2a+3b\right)$

2 a 2 + 5 ab + 3 b 2

$\left(r+s\right)\left(3r+2s\right)$

$\left(5x-y\right)\left(x-4\right)$

5 x 2 − 20 x xy + 4 y

$\left(4z-y\right)\left(z-6\right)$

Multiply a Trinomial by a Binomial

In the following exercises, using either method.

$\left(u+4\right)\left({u}^{2}+3u+2\right)$

u 3 + 7 u 2 + 14 u + 8

$\left(x+5\right)\left({x}^{2}+8x+3\right)$

$\left(a+10\right)\left(3{a}^{2}+a-5\right)$

3 a 3 + 31 a 2 + 5 a − 50

$\left(n+8\right)\left(4{n}^{2}+n-7\right)$

$\left(y-6\right)\left({y}^{2}-10y+9\right)$

y 3 − 16 y 2 + 69 y − 54

$\left(k-3\right)\left({k}^{2}-8k+7\right)$

$\left(2x+1\right)\left({x}^{2}-5x-6\right)$

2 x 3 − 9 x 2 − 17 x − 6

$\left(5v+1\right)\left({v}^{2}-6v-10\right)$

## Everyday math

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 $10+5.$

1. Multiply $\left(10+3\right)\left(10+5\right)$ by the FOIL method.
2. Multiply $13\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}15$ without using a calculator.
3. Which way is easier for you? Why?
1. 195
2. 195
3. Answers will vary.

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 $20-3.$

1. Multiply $\left(20-2\right)\left(20-3\right)$ by the FOIL method.
2. Multiply $18\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}17$ without using a calculator.
3. Which way is easier for you? Why?

## Writing exercises

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

Answers will vary.

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

## 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?

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