2. Multiplying and Dividing Rational
Expressions
Let a, b, c, and d be polynomials.
a c ac where
Algebra: × = b ¹ 0 and d ¹ 0
b d bd
a c a d ad where b ¹ 0, c ¹ 0, and d ¹ 0
¸ = × =
b d b c bc
Examples: x + 2 × 32 = 3(x + 2)
x x x3
x 4 x x x2
¸ = × =
x -1 x x -1 4 4(x -1)
3. Example 1 Multiply rational expressions involving monomials
2x 2 6x 2
Find the product • 4
.
3x 12x
2x 2 6x 2 (2x 2) (6x 2) Multiply numerators and
• 4
= denominators.
3x 12x (3x ) (12x 4 )
12x 4
= Product of powers property
36x 5
12 • x 4 Factor and divide out common
= factors.
3 • 12 • x 4 • x
1
= Simplify.
3x
4. Example 1 Multiply rational expressions involving monomials
CHECK Check your simplification using a graphing
calculator.
2x 2 6x 2
Graph y1 = • 4
and
3x 12x
1
y2 = .
3x
The graphs coincide. So, the expressions are
equivalent for all values of x other than 0.
5. Example 2 Multiple Choice Practice
x 2 – 4x + 3
3x 2 + 3x
What is the product of and ?
2 – x
4x 2 – 24x + 36 x
3x 3 ( x – 1) 3 ( x + 1) x+1
4 ( x – 3)
–
x 3 4 ( x + 3) x +3
SOLUTION
3x 2 + 3x x 2 – 4x + 3
•
4x 2 – 24x + 36 x2 – x
( 3x 2 + 3x ) ( x 2 – 4x + 3) Multiply numerators and
= denominators.
(4x 2 – 24x + 36)( x 2 – x)
6. Example 2 Multiple Choice Practice
3x ( x + 1) ( x – 3) ( x – 1) Factor and divide out
=
4x ( x – 3) ( x – 3) ( x – 1) common factors.
3 ( x + 1)
= Simplify.
4 ( x – 3)
ANSWER The correct answer is C.
7. Example 3 Multiply a rational expression by a polynomial
5x
Find the product 2 • ( x + 3).
x + 5x + 6
5x
2 + 5x
• ( x + 3)
x + 6
5x x+3 Rewrite polynomial as a
= 2 •
x + 5x + 6 1 fraction.
5x ( x + 3) Multiply numerators and
= 2 denominators.
x + 5x + 6
5x ( x + 3) Factor and divide out common
= factor.
( x + 2) ( x + 3)
8. Example 3 Multiply a rational expression by a polynomial
5x
= Simplify.
x+2
9. Example 4 Divide rational expressions involving polynomials
7x 2 – 7x x +1
Find the quotient 2 ÷ 2 .
x + 2x – 3 x – 7x – 8
7x 2 – 7x x +1
2 + 2x – 3
÷ 2
x x – 7x – 8
7x 2 – 7x x 2 – 7x – 8 Multiply by multiplicative
= 2 •
x + 2x – 3 x +1 inverse.
(7x 2 – 7x) ( x 2 – 7x – 8 ) Multiply numerators and
=
( x 2 + 2x – 3) ( x + 1) denominators.
7x ( x – 1) ( x – 8) ( x + 1) Factor and divide out
=
( x + 3) ( x – 1) ( x + 1) common factors.
10. Example 4 Divide rational expressions involving polynomials
7x ( x – 8) Simplify.
=
x +3
11. Example 5 Divide a rational expression by a polynomial
2x 2 + 16x + 24
Find the quotient 2 ÷ ( x + 6 ).
3x
2x 2 + 16x + 24
2 ÷ ( x + 6)
3x
2x 2 + 16x + 24 x +6 Rewrite polynomial as
= ÷
3x 2 1 fraction.
2x 2 + 16x + 24 1 Multiply by multiplicative
= • inverse.
3x 2 x +6
2x 2 + 16x + 24 Multiply numerators and
= denominators.
3x 2 ( x + 6 )
12. Example 5 Divide a rational expression by a polynomial
2 (x + 2) (x + 6) Factor and divide out
= common factor.
3x 2 ( x + 6 )
2 (x + 2)
= Simplify.
3x 2
13. Example 6 Solve a multi-step problem
BASEBALL
Hank Aaron’s career number B of times at bat and
career number H of hits through each year of the
period 1954–1976 can be modeled by
300 + 700x 62 + 240x
B = and H =
1 + 0.01x 1 + 0.017x
where x is the number of years since 1954. A baseball
player’s batting average is the number of hits divided
by the number of times at bat.
• Write a model that gives Hank Aaron’s career batting
average A as a function of x.
14. Example 6 Solve a multi-step problem
• Approximate his career batting average in 1959.
SOLUTION
STEP 1 Write a verbal model. Then write an equation.
= ÷
A = H ÷ B
STEP 2 Find the quotient.
A =H ÷B Write equation.
15. Example 6 Solve a multi-step problem
62 + 240x 300 + 700x
= ÷ Substitute for H and B.
1 + 0.017x 1 + 0.01x
62 + 240x 1 + 0.01x
= • Multiply by multiplicative
1 + 0.017x 300 + 700x inverse.
( 62 + 240x )( 1 + 0.01x ) Multiply numerators and
=
( 1 + 0.017x )( 300 + 700x ) denominators.
(2)( 31 + 120x )( 1 + 0.01x )
= Factor and divide out
( 1 + 0.017x ) (2)( 150 + 350x ) common factor.
16. Example 6 Solve a multi-step problem
( 31 + 120x )( 1 + 0.01x ) Simplify.
=
(1 + 0.017x ) (150 + 350x )
STEP 3 Approximate Aaron’s career batting average
in 1959. Substitute 5 for x in the model and use
a calculator to evaluate.
( 31 + 120 • 5 )( 1 + 0.01 • 5 )
A = ≈ .321
( 1 + 0.017 • 5)( 150 + 350 • 5 )
ANSWER
According to the model, Hank Aaron’s career batting
average in 1959 was approximately .321.
