1. Warm Up Lesson Presentation Lesson Quiz
Multiply and Divide Rational Expressions
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
Factor the polynomial.
1. x2 + 3x – 18
ANSWER
(x – 3)(x + 6)
2. 2x2 – 4x – 16
ANSWER
2(x – 4)(x + 2)
3. 2x2 + 8x – 10
ANSWER
2(x + 5)(x – 1)

3. Warm-Up 4.
A golf clinic charges a $30 enrollment fee and $5 per session. What is the average cost per session if you attend 8 sessions?
ANSWER
$8.75

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

5. Guided Practice
Find the product
2y3 5y
15y3
8y5 1. 3 4
ANSWER 2.
7z2
4z3 z3
14z
ANSWER z 8

6. Example 2 3x2 + 3x 4x2 – 24x + 36 x2 – 4x + 3 x2 – x Find the product=
(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.

7. Example 2
CHECK
Check your simplification using a graphing calculator.
Graph y1 =
3x2 + 3x
4x2 – 24x + 36
x2 – 4x + 3
x2 – x
and y2 =
3(x + 1)
4(x – 3) .
The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3).

8. Guided Practice 3.
Find the product
x2 + x – 2
x2 + x
2x2 + 2x
5x2 –15x +10
2(x + 1)
5(x – 2)
ANSWER

9. Example 3 Find the product 5x x2 + 5x + 6 (x + 3). 5x x2 + 5x + 6= 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.

10. Guided Practice
Find the product
2w2
w2 – 7w + 12
(w – 4). 4.
2w2
w – 3
ANSWER

11. Example 4 Find the quotient . 7x2 – 7x x2 + 2x – 3 x + 1 x2 – 7x – 8=
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.

12. Guided Practice
Find the quotient. 5.
m2 – 4
2m2 + 4m
6m – 3m2
4m + 44
2(m + 11)
3m2 –
ANSWER

13. Example 5 Find the quotient 2x2 + 16x + 24 3x2 (x + 6). 2x2 + 16x + 24=
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.

14. Guided Practice
Find the quotient. 6.
n2 – 6n + 9
12n
n – 3
ANSWER
12n
n – 3

15. Example 6
ADVERTISING
The amount A (in millions of dollars) spent on all advertising and the amount T (in millions of dollars) spent on television advertising in the United States during the period 1970–2003 can be modeled by
A =
13, x
1 – 0.015x
and
T = x
1 – 0.016x
where x is the number of years since Write a model that gives the percent p (in decimal form) of the amount spent on all advertising that was spent on television advertising. Then approximate the percent spent on television advertising in 2003.

16. Write a verbal model. Then write an equation.
Example 6
SOLUTION
STEP 1
Write a verbal model. Then write an equation. p = T A
STEP 2
Find the quotient.
p = T A
Write equation.
1800 +
860x 1 –
0.016
13,000
3700x
0.015x x =
Substitute for T and for A.

17. Example 6
1800 +
860x 1 –
0.016 x
13,000
3700x
1– 0.015x =
Multiply by multiplicative inverse.
( x)(1 – 0.015x)
(1– 0.016x)(13, x) =
Multiply numerators and denominators.
20 ( x) (1 – 0.015x)
(1 – 0.016x)(20)( x) =
Factor and divide out common factor.
( x)(1 – 0.015x)
(1 – 0.016x)( x) =
Simplify.

18. Example 6
STEP 3
Approximate the percent spent on television advertising in 2003.Because 2003 – 1970 = 33, x = 33. Substitute 33 for x in the model and use a calculator to evaluate.
( )(1 – )
(1 – )( )
P =
0.239
ANSWER
About 24% of the amount spent on all advertising was spent on television advertising in 2003.

19. Guided Practice 7.
In Example 6, find the values of T and of A separately when x = 33. Then divide the value of T by the value of A. Compare your answer with the answer in Step 3 above.
ANSWER
About $63,941 million
About $267,525 million
About 0.239
The answers are the same.

20. Lesson Quiz
Find the product 3x
x2 – 3x – 10
(x + 2)
ANSWER 3x
x – 5
2. Find the quotient
( 2x3 + 11x2 + 11x – 4 ) ( x + 4)
ANSWER
2x2 + 3x – 1

21. Lesson Quiz 3.
The average total revenue (in thousands of dollars) of books B sold at a used bookstore and the average revenue of paper back books P sold at the store during 1990 – 2004 can be represented by and where x is the number of years since Approximate the percent of the total revenue of the books from paperback sales in 2004. x
1 – 0.02x
B =
720 – 36x
1 – 0.017x
P =
ANSWER
About 21%