# 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.

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
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5%
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Tickets for the community fair cost $12 for adults and$5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold? Alpha Reply 220 gayla Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert? Tsimmuaj Reply Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj ? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year candido what would be wong candido 87 divided by 3 then multiply that by 4. 116 people total. Melissa the actual number that has 3 out of 4 of a whole pie candido was having a hard time finding Teddy use Matrices for the 2nd question Daniel One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers. Tsimmuaj Reply 26 + 37 = 63 + 8 = 71 gayla 26+37=63+8=71 ziad 11+52=63+8=71 Thisha how do we know the answer is correct? Thisha 23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for. gayla 23 +11 = 37. 23+37=63 63+8=71 Gayla by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71 Gayla your answer did not fit the guidelines of the question 11 is 41 less than 52. gayla 71-8-11 =52 is this correct? Ruel let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer tesfu just new Muwanga Amara currently sells televisions for company A at a salary of$17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of$29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal? Tsimmuaj Reply yes math Kenneth company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100 gayla need help with math to do tsi test Toocute me too Christian have you tried the TSI practice test ***tsipracticetest.com gayla DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players. gustavo Reply ? Ann DaMarcus: 16 mi/hr Fabian: 22 mi/hr Sherman Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution? Wenda Reply 15 and 5 32 is 40% , & 8 is 10 % , & any 4 letters is 5%. Karen It felt that something is missing on the question like: 40% of what solution? 10% of what solution? Jhea its confusing Sparcast 3% & 2% to complete the 25% Sparcast because she already has 20 liters. Sparcast ok I was a little confused I agree 15% & 5% Sparcast 8,2 Karen Jim and Debbie earned$7200. Debbie earned \$1600 more than Jim earned. How much did they earned
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