Here are the steps to divide the polynomials: 1) (a^2 + 3a - 4) ÷ (a - 1) = a + 4 2) (9b^2 + 6b + 8) ÷ (3b - 4) = 3b + 2 3) (8m - 7 + 4m^2) ÷ (5 + 2m) = 2m - 1

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
=
16x 8
1 x5 1
4x 5
4x5 12 12x5
4x3 4 x3
= • 8 Rewrite using product rule for
16x 8
16 x fractions.
1 Quotient of powers property
= • x– 5
4
1 1 Definition of negative exponents
= • 5
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.
?
( 2x2 + 4x + 5) = 4x3 + 8x2 + 10x
2x
?
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 ÷ 1+
3 = Which is really 1
3 3
12
(2 x + 11x − 9) ÷ ( 2 x − 3) = x + 7 +
2
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 (2x + 11x – 9) ÷ ( 2x – 3) = x + 7 +
2
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 + y + 4) ÷ ( 2 + y) = y + 3 +
2
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 + 4m ) ÷ (–1 + 2m) = 2m + 1 +
2
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 + 3 x 2 − 12 x) ÷ 3 x

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)