Dividing polynomials involves writing the division as a fraction and using properties of exponents. Terms are divided using the quotient of powers property. Any remaining terms are the remainder. Examples show dividing monomials, polynomials by monomials and binomials, checking work, and rewriting polynomials with missing terms or nonzero remainders.

2. Dividing by a Monomial
 Write the division as a fraction and use the quotient of
powers property.
 When dividing polynomials, you can check your work
using multiplication.

3. Example 1 Divide monomials
Divide –8x5 by 2x2.
SOLUTION
Write the division as a fraction and use the quotient
of powers property.
–8x5
–8x5 ÷ (2x2) = Write as fraction.
2x2
– 8 x5 Rewrite using product rule for
= • 2 fractions.
2 x
–8
= • x5 – 2 Quotient of powers property
2

4. Example 1 Divide monomials
= –4x3 Simplify.

5. Example 2 Multiple Choice Practice
4x3
8
=
16x
1 x5 1
4x5
4x5 12 12x5
4x3 4 x3
8
= • 8 Rewrite using product rule for fractions.
16x 16 x
1
= • x–5 Quotient of powers property
4
1 1
= • 5 Definition of negative exponents
4 x

6. Example 2 Multiple Choice Practice
1 Simplify.
= 5
4x
ANSWER The correct answer is B.

7. Example 3 Divide a polynomial by a monomial
Divide 4x3 + 8x2 + 10x by 2x.
SOLUTION
4x3 + 8x2 + 10x
( 4x3 + 8x2 + 10x ) ÷ 2x = Write as fraction.
2x
4x3 8x2 10x
= + + Divide each term
2x 2x 2x by 2x.
= 2x2 + 4x + 5 Simplify.

8. Example 3 Divide a polynomial by a monomial
CHECK Check to see if the product of 2x and 2x2 + 4x + 5
is 4x3 + 8x2 + 10x.
?
2x ( 2x2 + 4x + 5) = 4x3 + 8x2 + 10x
?
2x ( 2x2) + 2x (4x ) + 2x (5 ) = 4x3 + 8x2 + 10x
4x3 + 8x2 + 10x = 4x3 + 8x2 + 10x

9. Division with Algebra Tiles
 Pg. 540

10. Dividing by a Binomial
 To divide a polynomial by a binomial, use long
division.

11. Example 4 Divide a polynomial by a binomial
Divide x2 + 2x – 3 by x – 1.
SOLUTION
STEP 1 Divide the first term of x2 + 2x – 3 by the first
term of x – 1.
x
x – 1 x2 + 2x – 3 Think: x2 ÷ x = ?
x2 – x Multiply x – 1 by x.
3x Subtract x2 – x from x2 + 2x.

12. Example 4 Divide a polynomial by a binomial
STEP 2 Bring down –3. Then divide the first term of
3x – 3 by the first term of x – 1.
x + 3
x – 1 x2 + 2x – 3
x2 – x
3x – 3 Think: 3x ÷ x = ?
3x – 3 Multiply x – 1 by 3.
0 Subtract 3x – 3 from 3x – 3;
remainder is 0.
ANSWER ( x2 + 2x – 3) ÷ (x – 1) = x + 3

13. Nonzero Remainders
 When you obtain a nonzero remainder, apply the
following rule:
Re mainder
Dividend Divisor Quotient
Divisor
2 2
5 3 1 Which is really 1
3 3
2 12
(2 x 11x 9) (2 x 3) x 7
2x 3

14. Example 5 Divide a polynomial by a binomial
Divide 2x2 + 11x – 9 by 2x – 3.
x + 7
2x – 3 2x2 + 11x – 9
2x2 – 3x Multiply 2x – 3 by x.
14x – 9 Subtract 2x2 – 3x. Bring down – 9.
14x – 21 Multiply 2x – 3 by 7.
12 Subtract 14x – 21; remainder is 12.
12
ANSWER (2x2 + 11x – 9) ÷ ( 2x – 3) = x + 7 +
2x – 3

15. Example 6 Rewrite polynomials
Divide 5y + y2 + 4 by 2 + y.
y + 3
y + 2 y2 + 5y + 4 Rewrite polynomials.
y2 + 2y Multiply y + 2 by y.
3y + 4 Subtract y2 + 2y. Bring down 4.
3y + 6 Multiply y + 2 by 3.
–2 Subtract 3y + 6; remainder is – 2.
–2
ANSWER (5y + y2 + 4) ÷ ( 2 + y) = y + 3 +
y +2

16. Example 7 Insert missing terms
Divide 13 + 4m2 by –1 + 2m.
2m + 1 Rewrite polynomials. Insert
2m – 1 4m2 + 0m + 13 missing term.
4m2 – 2m Multiply 2m – 1 by 2m.
2m + 13 Subtract 4m2 – 2m. Bring down 13.
2m – 1 Multiply 2m – 1 by 1.
14 Subtract 2m – 1; remainder is 14.
14
ANSWER (13 + 4m2) ÷ (–1 + 2m) = 2m + 1 +
2m – 1

17. 9.4 Warm-Up (Day 1)
 Divide.
1. 3d 7 ( 9d 4 )
2. 8 z ( 6 z 5 )
3. (6 x 3 3x 2 12 x) 3x

18. 9.4 Warm-Up (Day 2)
 Divide.
1. (a 2 3a 4) (a 1)
2. (9b 2 6b 8) (3b 4)
3. (8m 7 4m 2 ) (5 2m)