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By the end of this section, you will be able to:
  • Divide a polynomial by a monomial
  • Divide a polynomial by a binomial

Before you get started, take this readiness quiz.

  1. Add: 3 d + x d .
    If you missed this problem, review [link] .
  2. Simplify: 30 x y 3 5 x y .
    If you missed this problem, review [link] .
  3. Combine like terms: 8 a 2 + 12 a + 1 + 3 a 2 5 a + 4 .
    If you missed this problem, review [link] .

Divide a polynomial by a monomial

In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial    .

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.

The sum, y 5 + 2 5 , simplifies to y + 2 5 .

Now we will do this in reverse to split a single fraction into separate fractions.

We’ll state the fraction addition property here just as you learned it and in reverse.

Fraction addition

If a , b , and c are numbers where c 0 , then

a c + b c = a + b c and a + b c = a c + b c

We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.

For example, y + 2 5 can be written y 5 + 2 5 .

We use this form of fraction addition to divide polynomials by monomials.

Division of a polynomial by a monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial    .

Find the quotient: 7 y 2 + 21 7 .

Solution

7 y 2 + 21 7 Divide each term of the numerator by the denominator. 7 y 2 7 + 21 7 Simplify each fraction. y 2 + 3

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Find the quotient: 8 z 2 + 24 4 .

2 z 2 + 6

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Find the quotient: 18 z 2 27 9 .

2 z 2 3

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Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.

Find the quotient: ( 18 x 3 36 x 2 ) ÷ 6 x .

Solution

( 18 x 3 36 x 2 ) ÷ 6 x Rewrite as a fraction. 18 x 3 36 x 2 6 x Divide each term of the numerator by the denominator. 18 x 3 6 x 36 x 2 6 x Simplify. 3 x 2 6 x

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Find the quotient: ( 27 b 3 33 b 2 ) ÷ 3 b .

9 b 2 11 b

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Find the quotient: ( 25 y 3 55 y 2 ) ÷ 5 y .

5 y 2 11 y

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When we divide by a negative, we must be extra careful with the signs.

Find the quotient: 12 d 2 16 d −4 .

Solution

12 d 2 16 d −4 Divide each term of the numerator by the denominator. 12 d 2 −4 16 d −4 Simplify. Remember, subtracting a negative is like adding a positive! −3 d 2 + 4 d

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Find the quotient: 25 y 2 15 y −5 .

−5 y 2 + 3 y

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Find the quotient: 42 b 2 18 b −6 .

−7 b 2 + 3 b

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Find the quotient: 105 y 5 + 75 y 3 5 y 2 .

Solution

105 y 5 + 75 y 3 5 y 2 Separate the terms. 105 y 5 5 y 2 + 75 y 3 5 y 2 Simplify. 21 y 3 + 15 y

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Find the quotient: 60 d 7 + 24 d 5 4 d 3 .

15 d 4 + 6 d 2

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Find the quotient: 216 p 7 48 p 5 6 p 3 .

36 p 4 8 p 2

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Find the quotient: ( 15 x 3 y 35 x y 2 ) ÷ ( −5 x y ) .

Solution

( 15 x 3 y 35 x y 2 ) ÷ ( −5 x y ) Rewrite as a fraction. 15 x 3 y 35 x y 2 −5 x y Separate the terms. Be careful with the signs! 15 x 3 y −5 x y 35 x y 2 −5 x y Simplify. −3 x 2 + 7 y

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Find the quotient: ( 32 a 2 b 16 a b 2 ) ÷ ( −8 a b ) .

−4 a + 2 b

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Find the quotient: ( −48 a 8 b 4 36 a 6 b 5 ) ÷ ( −6 a 3 b 3 ) .

8 a 5 b + 6 a 3 b 2

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Find the quotient: 36 x 3 y 2 + 27 x 2 y 2 9 x 2 y 3 9 x 2 y .

Solution

36 x 3 y 2 + 27 x 2 y 2 9 x 2 y 3 9 x 2 y Separate the terms. 36 x 3 y 2 9 x 2 y + 27 x 2 y 2 9 x 2 y 9 x 2 y 3 9 x 2 y Simplify. 4 x y + 3 y y 2

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Find the quotient: 40 x 3 y 2 + 24 x 2 y 2 16 x 2 y 3 8 x 2 y .

5 x y + 3 y 2 y 2

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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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