1. 11.5 Multiplying and Dividing Rational Expressions
Multiply and divide rational expressions.
Rules for multiplying and dividing rational expressions are the same as for multiplying and dividing fractions.

2. 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.

3. 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
Multiply numerators and denominators. =
(3x2 + 3x) (x2 – 4x + 3)
(4x2 – 24x + 36)(x2 – x)
Factor and divide out common factors. =
3x(x + 1)(x – 3)(x – 1)
4(x2 – 6x + 9)x(x – 1) =
3x(x + 1)(x – 3)(x – 1)
4x(x – 3)(x – 3)(x – 1) =
3(x + 1)
4(x – 3)
Simplify.

4. 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).
Rewrite polynomial as a fraction. 5x
x2 + 5x + 6
(x + 3) = 1
Multiply numerators and denominators. =
5x(x + 3)
x2 + 5x + 6
Factor and divide out common factor. =
5x(x + 3)
(x + 3)
(x + 2) = 5x
x + 2
Simplify.

5. 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
Multiply by multiplicative inverse.
7x2 – 7x
x2 + 2x – 3
x + 1 =
x2 – 7x – 8
Multiply numerators and denominators. =
(7x2 – 7x)
(x2 – 7x – 8)
(x + 1)
(x2 + 2x – 3)
Factor and divide out common factors. =
7x(x – 1)(x – 8)(x + 1)
(x + 3)(x – 1)(x + 1)
Simplify. =
7x(x – 8)
x + 3

6. 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).
Rewrite polynomial as fraction.
2x2 + 16x + 24
3x2
(x +6). = 1
Multiply by multiplicative inverse.
2x2 + 16x + 24
3x2 = 1
(x +6).
Multiply numerators and denominators.
2x2 + 16x + 24
3x2 =
(x +6).
Factor and divide out common factor. =
2(x + 2)(x + 6)
3x2(x + 6) =
2(x + 2)
3x2
Simplify.

7. TRY THESE
Find the product
2y3 5y
15y3
8y5 1. 3 4
ANSWER
ANSWER z 8 2.
7z2
4z3 z3
14z
x2 + x – 2
x2 + x
2x2 + 2x
5x2 –15x +10 3.
2(x + 1)
5(x – 2)
ANSWER
2w2
w2 – 7w + 12
(w – 4). 4.
2w2
w – 3
ANSWER

8. 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