1. Do Now Pass out calculators. Have your homework out ready to check.
Solve the problem below:
Divide y2 by y.

2. Objective:
To multiply and divide rational expressions.

3. Multiply rational expressions involving monomials
EXAMPLE 1
Multiply rational expressions involving monomials
Find the product
2x2 3x
6x2
12x3
2x2 3x
6x2
12x3 =
(2x2)(6x2)
(3x)(12x3)
Multiply numerators and denominators.
12x4
36x4 =
Product of powers property
12 x4 x4 =
Factor and divide out common factors. 1 3 =
Simplify.

4. GUIDED PRACTICE
for Example 1
Find the product
2y3 5y
15y3
8y5 1. 3 4
ANSWER 2.
7z2
4z3 z3
14z
ANSWER z 8

5. Multiply rational expressions involving polynomials
EXAMPLE 2
Multiply rational expressions involving polynomials
3x2 + 3x
4x2 – 24x + 36
x2 – 4x + 3
x2 – x
Find the product
3x2 + 3x
4x2 – 24x + 36
x2 – 4x + 3
x2 – x =
(3x2 + 3x) (x2 – 4x + 3)
(4x2 – 24x + 36)(x2 – x)
Multiply numerators and denominators. =
3x(x + 1)(x – 3)(x – 1)
4x(x – 3)(x – 3)(x – 1)
Factor and divide out common factors. =
3(x + 1)
4(x – 3)
Simplify.

6. Multiply a rational expression by a polynomial
EXAMPLE 3
Multiply a rational expression by a polynomial
Find the product 5x
x2 + 5x + 6
(x + 3). 5x
x2 + 5x + 6
(x + 3). 5x
x2 + 5x + 6
(x + 3) = 1
Rewrite polynomial as a fraction. =
5x(x + 3)
x2 + 5x + 6
Multiply numerators and denominators. =
5x(x + 3)
(x + 3)
(x + 2)
Factor and divide out common factor. = 5x
x + 2
Simplify.

7. GUIDED PRACTICE
for Examples 2 and 3
x2 + x – 2
x2 + x
2x2 + 2x
5x2 –15x +10 3.
Find the product
2(x + 1)
5(x – 2)
ANSWER
Find the product
2w2
w2 – 7w + 12
(w – 4). 4.
2w2
w – 3
ANSWER

8. Divide rational expressions involving polynomials
EXAMPLE 4
Divide rational expressions involving polynomials
Find the quotient
7x2 – 7x
x2 + 2x – 3
x + 1
x2 – 7x – 8
7x2 – 7x
x2 + 2x – 3
x + 1
x2 – 7x – 8
7x2 – 7x
x2 + 2x – 3
x + 1 =
x2 – 7x – 8
Multiply by multiplicative inverse. =
(7x2 – 7x)
(x2 – 7x – 8)
(x + 1)
(x2 + 2x – 3)
Multiply numerators and denominators. =
7x(x – 1)(x – 8)(x + 1)
(x + 3)(x – 1)(x + 1)
Factor and divide out common factors. =
7x(x – 8)
x + 3
Simplify.

9. Divide a rational expression by a polynomial
EXAMPLE 5
Divide a rational expression by a polynomial
Find the quotient
2x2 + 16x + 24
3x2
(x +6).
2x2 + 16x + 24
3x2
(x +6) =
2x2 + 16x + 24
3x2
(x +6). = 1
Rewrite polynomial as fraction.
2x2 + 16x + 24
3x2 = 1
(x +6).
Multiply by multiplicative inverse.
2x2 + 16x + 24
3x2 =
(x +6).
Multiply numerators and denominators. =
2(x + 2)(x + 6)
3x2(x + 6)
Factor and divide out common factor. =
2(x + 2)
3x2
Simplify.

10. GUIDED PRACTICE
for Examples 4 and 5
Find the quotient. 5.
m2 – 4
2m2 + 4m
6m – 3m2
4m + 44
2 (m +11)
3m2 –
ANSWER 6.
n2 – 6n + 9
12n
n – 3
ANSWER
12n
n – 3