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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 ( x + 3 ) .
    If you missed this problem, review [link] .
  2. Combine like terms: x 2 + 9 x + 7 x + 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 ( x − 3 ) . You multiplied both terms in the parentheses, x and 3 , by 2, to get 2 x − 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 ( x + 3 ) .

Solution

4 times x plus 3. Two arrows extend from 4, terminating at x and 3.
Distribute. 4 times x plus 4 times 3.
Simplify. 4 x plus 12.

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Multiply: 5 ( x + 7 ) .

5 x + 35

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Multiply: 3 ( y + 13 ) .

3 y + 39

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Multiply: y ( y − 2 ) .

Solution

y times y minus 2. Two arrows extend from the coefficient y, terminating at the y and minus 2 in parentheses.
Distribute. y times y minus y times 2.
Simplify. y squared minus 2 y.

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Multiply: x ( x − 7 ) .

x 2 − 7 x

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Multiply: d ( d − 11 ) .

d 2 − 11 d

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Multiply: 7 x ( 2 x + y ) .

Solution

7 x times 2 x plus y. Two arrows extend from 7x, terminating at 2x and y.
Distribute. 7 x times 2 x plus 7 x times y.
Simplify. 14 x squared plus 7 x y.

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Multiply: 5 x ( x + 4 y ) .

5 x 2 + 20 x y

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Multiply: 2 p ( 6 p + r ) .

12 p 2 + 2 p r

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Multiply: −2 y ( 4 y 2 + 3 y − 5 ) .

Solution

Negative 2 y times 4 y squared plus 3 y minus 5. Three arrows extend from negative 2 y, terminating at 4 y squared, 3 y, and minus 5.
Distribute. Negative 2 y times 4 y squared plus negative 2 y times 3 y minus negative 2 y times 5.
Simplify. Negative 8 y cubed minus 6 y squared plus 10 y.

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Multiply: −3 y ( 5 y 2 + 8 y − 7 ) .

−15 y 3 − 24 y 2 + 21 y

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Multiply: 4 x 2 ( 2 x 2 − 3 x + 5 ) .

8 x 4 − 24 x 3 + 20 x 2

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Multiply: 2 x 3 ( x 2 − 8 x + 1 ) .

Solution

2 x cubed times x squared minus 8 x plus 1. Three arrows extend from 2 x cubed, terminating at x squared, minus 8 x, and 1.
Distribute. 2 x cubed times x squared plus 2 x cubed times negative 8 x plus 2 x cubed times 1.
Simplify. 2 x to the fifth power minus 16 x to the fourth power plus 2 x cubed.

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Multiply: 4 x ( 3 x 2 − 5 x + 3 ) .

12 x 3 − 20 x 2 + 12 x

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Multiply: −6 a 3 ( 3 a 2 − 2 a + 6 ) .

−18 a 5 + 12 a 4 − 36 a 3

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Multiply: ( x + 3 ) p .

Solution

The monomial is the second factor. x plus 3, in parentheses, times p. Two arrows extend from the p, terminating at x and 3.
Distribute. x times p plus 3 times p.
Simplify. x p plus 3 p.

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Multiply: ( x + 8 ) p .

x p + 8 p

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Multiply: ( a + 4 ) p .

a p + 4 p

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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    .

x plus 3, in parentheses, times p. Two arrows extend from the p, terminating at x and 3.
We distributed the p to get: x p plus 3 p.
What if we have ( x + 7) instead of p ? x plus 3 multiplied by x plus 7. Two arrows extend from x plus 7, terminating at the x and the 3 in the first binomial.
Distribute ( x + 7). The sum of two products. The product of x and x plus 7, plus the product of 3 and x plus 7.
Distribute again. x squared plus 7 x plus 3 x plus 21.
Combine like terms. x squared plus 10 x plus 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.

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

Solution

The product of two binomials, y plus 5 and y plus 8. Two arrows extend from y plus 8, terminating at the y and the 5 in the first binomial.
Distribute ( y + 8). The sum of two products, the product of y and y plus 8, plus the product of 5 and y plus 8.
Distribute again y squared plus 8 y plus 5 y plus 40.
Combine like terms. y squared plus 13 y plus 40.

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Multiply: ( x + 8 ) ( x + 9 ) .

x 2 + 17 x + 72

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Multiply: ( 5 x + 9 ) ( 4 x + 3 ) .

20 x 2 + 51 x + 27

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

Solution

The product of two binomials, 2 y plus 5 and 3 y plus 4. Two arrows extend from 3y plus 4, terminating at 2y and 5 in the first binomial.
Distribute (3 y + 4). The sum of two products, the product of 2 y and 3 y plus 4, plus the product of 5 and 3 y plus 4.
Distribute again 6 y squared plus 8 y plus 15 y plus 20.
Combine like terms. 6 y squared plus 23 y plus 20.

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Multiply: ( 3 b + 5 ) ( 4 b + 6 ) .

12 b 2 + 38 b + 30

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Multiply: ( a + 10 ) ( a + 7 ) .

a 2 + 17 a + 70

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Multiply: ( 4 y + 3 ) ( 2 y − 5 ) .

Solution

The product of two binomials, 4y plus 3 and 2 y minus 5. Two arrows extend from 2y minus 5, terminating at 4 y and 3 in the first binomial.
Distribute. The sum of two products, the product of 4y and 2y minus 5, plus the product of 3 and 2y minus 5.
Distribute again. 8 y squared minus 20 y plus 6 y minus 15.
Combine like terms. 18 y squared minus 14 y minus 15.

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Multiply: ( 5 y + 2 ) ( 6 y − 3 ) .

30 y 2 − 3 y − 6

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Multiply: ( 3 c + 4 ) ( 5 c − 2 ) .

15 c 2 + 14 c − 8

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Multiply: ( x + 2 ) ( x − y ) .

Solution

The product of two binomials, x minus 2 and x minus y. Two arrows extend from x minus y, terminating at x and 2 in the first binomial.
Distribute. The difference of two products. The product of x and x minus 7, minus the product of 2 and x minus y.
Distribute again. x squared minus x y minus 2 x plus 2 y.
There are no like terms to combine.

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Multiply: ( a + 7 ) ( a − b ) .

a 2 − a b + 7 a − 7 b

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Multiply: ( x + 5 ) ( x − y ) .

x 2 − x y + 5 x − 5 y

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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.

( x − 2 ) ( x − y ) x 2 − x y − 2 x + 2 y

Where did the first term, x 2 , come from?

This figure explains how to multiply a binomial using the FOIL method. It has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says “It is the product of x and x, the first terms in x minus 2 and x minus y.” In the right column is the product of x minus 2 and x minus y. An arrow extends from the x in x minus 2, and terminates at the x in x minus y. Below this is the word “First.” One row down, the text in the left column says “The next terms, negative xy, is the product of x and negative y, the two outer terms.” In the right column is the product of x minus 2 and x minus y, with another arrow extending from the x in x minus 2 to the y in x minus y. Below this is the word “Outer.” One row down, the text in the left column says “The third term, negative 2 x, is the product of negative 2 and x, the two inner terms.” In the right column is the product of x minus 2 and x minus y with a third arrow extending from minus 2 in x minus 2 and terminating at the x in x minus y. Below this is the word “Inner.” In the last row, the text in the left column says “And the last term, plus 2y, came from multiplying the two last terms, negative 2 and negative y.” In the right column is the product of x minus 2 and x minus y, with a fourth arrow extending from the minus 2 in x minus 2 to the minus y in x minus y. Below this is the word “Last.”

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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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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