12-4
Multiplying and Dividing Rational Expressions
12-4
1. Plan
Lesson Preview What You’ll Learn
Check Skills You’ll Need
(For help, go to Lessons 8-3 and 9-6.)
OBJECTIVE
To multiply rational expressions
1 OBJECTIVE
To divide rational expressions
2
. . . And Why
Simplify each expression. 10 1. r 2 ? r 8 r
2. b 3 ? b 4 b7
3. c 7 4 c 2 c5
4. 3x4 ? 2x5 6x9
5. 5n2 ? n2 5n4
6. 15a3(-3a2) –45a5
Factor each polynomial. 7. 2c 2 + 15c + 7 (2c ± 1)(c ± 7)
To find loan payments, as in Exercises 38–40
8. 15t 2 - 26t + 11 (15t – 11)(t – 1)
9. 2q 2 + 11q + 5 (2q ± 1)(q ± 5)
Interactive lesson includes instant self-check, tutorials, and activities.
OBJECTIVE
1
Part 1 1 Multiplying Rational Expressions
Multiplying rational expressions is similar to multiplying rational numbers. ac If a, b, c, and d represent polynomials (with b 2 0 and d 2 0), then ba ? dc = bd .
Need Help?
1
Remember that the value of the expression in the denominator cannot be zero.
Check Understanding
Lesson Preview
✓Check Skills You’ll Need Multiplication Properties of Exponents Lesson 8-3: Example 2 Exercises 7–9 Extra Practice, p. 709
Factoring Trinomials of the Type ax2 ± bx ± c Lesson 9-6: Example 1 Exercises 1–9 Extra Practice, p. 710
Lesson Resources
Multiplying Rational Expressions
EXAMPLE
Teaching Resources Practice, Reteaching, Enrichment
Multiply. a. 3 ? 42
x x 3 ? 4 = 12 x x2 x3 x 3 x b. x 1 4 ? x 2 22 x(x 2 3) x x23 x 1 4 ? x 2 2 = (x 1 4)(x 2 2)
Reaching All Students Practice Workbook 12-4 Spanish Practice Workbook 12-4 Basic Algebra Planning Guide 12-4
Multiply the numerators and multiply the denominators.
Multiply the numerators and multiply the denominators. Leave the answer in factored form.
1 Multiply.
a. 62 ? 22 3 a
a
2 5 x 2 7 (x 2 5)(x 2 7) b. x x(x 1 3) x13? x
212 a5
ac As with rational numbers, the product bd may not be in simplest form. Look for factors common to the numerator and the denominator to divide out.
2
EXAMPLE
Using Factoring
6x . 3 4x 2 2 1 2x 1 1 6x 2x 1 1 ? 26x = ?( 2x 1 1)( 2x 2 1) 4x 2 1 3 3
Presentation Assistant Plus! Transparencies • Check Skills You’ll Need 12-4 • Additional Examples 12-4 • Student Edition Answers 12-4 • Lesson Quiz 12-4 PH Presentation Pro CD 12-4
Computer Test Generator CD
Multiply 2x 1 1 and
1 = 2x 1 1 ? 13
1 (2x
62x 1 1)(2x 2 1)
= 2x2x 21
Check Understanding
Technology Resource Pro® CD-ROM Computer Test Generator CD Prentice Hall Presentation Pro CD
Factor the denominator. Divide out the common factors 3 and (2x ± 1). Simplify.
www.PHSchool.com
2 16. –1 2 Multiply x 2 2 and 28x x 2 8x x 24
Lesson 12-4 Multiplying and Dividing Rational Expressions
657
✓ Ongoing Assessment and Intervention Before the Lesson
During the Lesson
After the Lesson
Diagnose prerequisite skills using: • Check Skills You’ll Need
Monitor progress using: • Check Understanding • Additional Examples • Standardized Test Prep
Assess knowledge using: • Lesson Quiz • Computer Test Generator CD
Student Site • Teacher Web Code: aek-5500 • Self-grading Lesson Quiz Teacher Center • Lesson Planner • Resources Plus
657
2. Teach
You can also multiply a rational expression by a polynomial. Leave the product in factored form.
Math Background
3
In addition to excluding values of the variable that make the function undefined, eliminating values that do not apply is particularly important in application problems.
Multiplying a Rational Expression by a Polynomial
EXAMPLE
12 2 Multiply 3s 2s 1 4 and s + 5s + 6. (s 1 2)(s 1 3) 3s 1 2 3s 1 2 2 2s 1 4 ? (s + 5s + 6) = 2(s 1 2) ? 1 1 1 2 ? (s 1 2) (s 1 3) = 2 3s 1 1 (s 1 2)
=
OBJECTIVE
1
Teaching Notes Additional Examples
Check Understanding
3 Multiply. a. 3c ? A c 3 - c B
c. (m - 1) ? 4m2 1 8 m 21
4(m 1 2) m 1 1
Recall that ba 4 dc = ba ? dc, where b 2 0, c 2 0, and d 2 0. When you divide rational expressions that can be factored, first rewrite the expression using the reciprocal before dividing out common factors.
y y x(x 2 2) x 2 2 x x 1 5 ? x 2 6 (x 1 5)(x 2 6)
2 Multiply 3x 41 1 and 2x 3x 2 1
8x . 9x2 2 1
Reading Math
5x 1 1 3 Multiply 3x 1 12 and (5x 1 1)(x 1 3) x2 + 7x + 12. 3
4
a26
a2
Additional Examples 4 Divide (x ± 5)(x ± 7)
by
x 1 8 . x2 2 49
=
(a 1 2)(a 1 5) (a 2 6)(a 1 6) ? a15 (a 2 6)
=
(a 1 2)(a 1 5) 1 (a 2 6) 1 (a 1 6) ? 1a 1 5 1 (a 2 6)
= (a + 2)(a + 6)
Check Understanding
224a22 a. a ab a
a
5
the reciprocal of a2 1 5 . a 2 36
Factor.
Divide out the common factors a ± 5 and a – 6. Leave in factored form.
88 x
5
2n 2 n 2 3
5(7m 2 10) 7(m 2 10)
The reciprocal of a polynomial such as 5x 2 + 5x is
(8x2 + 16x). x 1 27
2 c. 6n 2 2 5n 2 6 4 2n 2 3
5m 1 10 4 7m 1 14 b. 2m 2 20 14m 2 20
1 b
Closure
2 36, Multiply by a 2 1
4 Divide.
2 9x 1 14 5 Divide x 1 11x by
Ask students to explain how to multiply and divide rational expressions. To multiply rational expressions, factor each numerator and denominator, divide the numerators and denominators by any common factors, then multiply the remaining factors of the numerators and denominators. To divide rational expressions, multiply the dividend by the reciprocal of the divisor and proceed as above.
a 2 36
1 7a 1 10 4 a 1 5 = a 2 1 7a 1 10 ? a 2 2 36 a26 a 2 2 36 a26 a15
Teaching Notes
x2 1 13x 1 40 x 2 7
Dividing Rational Expressions
EXAMPLE
2 Divide a 1 7a 1 10 by a2 1 5 .
The vinculum or fraction bar is a grouping symbol.
OBJECTIVE
658
Leave in factored form.
2
a. 7y ? 82 563
2
(3s 1 2)(s 1 3) 2 2 b. v 2v 1 3 ? A v - 2v - 15 B 2v(v – 5)
3(c – 1)(c ± 1)
OBJECTIVE
Divide out the common factor s ± 2.
Part 1 2 Dividing Rational Expressions
1 Multiply.
b.
Factor.
EXAMPLE
1 . 5x 2 1 5x
n11
3n 1 2 2n 2 3
Dividing a Rational Expression by a Polynomial
2 3x 1 2 Divide x 1 4x by A 5x 2 + 5x B .
Multiply by the reciprocal of 5x 2 ± 5x.
x 2 1 3x 1 2 4 5x 2 1 5x = x 2 1 3x 1 2 ? 1 4x 1 4x 5x 2 1 5x (x 1 1)(x 1 2) = ? 5x(x11 1) 4x
=
(x 1
1) 1 (x 4x
1 2)
? 5x (x1 1 1) 1
= x 1 22 20x
658
Factor. Divide out the common factor x ± 1. Simplify.
Chapter 12 Rational Expressions and Functions
Reaching All Students Below Level Caution students that only common factors of the entire numerator and denominator can be divided out. x2 2 y2 For example, x 2 y 2 x - y.
Advanced Learners Have students
Error Prevention
simplify ¢ x 1 4 ? x 2 3 ≤ 3 x 12 4 . x11 x22 x 2 x 2 2x 2 x – 3x
See note on page 659.
3. Practice
5 Divide.
3 5 a. 3x 2 4 A-15x B
2 c. z2 1 2z 2 15 4 (z - 3)
y13
b. y 1 2 4 (y + 2)
– 12
z 1 9z 1 20
y 1 3 (y 1 2) 2
10x
EXERCISES
1 z 1 4
Assignment Guide
1
For more practice, see Extra Practice.
Practiceand andProblem ProblemSolving Solving Practice m(m 2 2)
A
Practice by Example Example 1 (page 657)
Example 2 (page 657)
Example 3 (page 658)
Example 4 (page 658)
35x 1. 37 ? 5x 12 36
12. x 2 1
m 2 2 ? 2m 1 6 1 10. 3m 1 9 2m 2 4 3
3a a 3a 2 2x(x 2 1) 2x x 2 1 6x 2 12x 2 5. x 1 1 ? 3 3(x 1 1) 6. 5 ? x 1 1 5(x 1 1) 3 4 4 6 9 8. 5x2 ? 3x 5x2 9. 3t ? 3t 2 t t22 6x x t2 x 2 5 ? 6x 1 9 1 4x 1 1 ? 30x 1 60 11. 4x 12. 5x 2x 2 2 1 6 3x 2 15 2 1 10
14 2 13. 4t t 2 3 ? At - t - 6B
11 A 2 B 14. 2m 3m 2 6 ? 9m - 36
26d 2 –2d 2 5 17. 2d 25 6d 2
18. c 2 - 1
2
Example 5 (page 658)
B
Apply Your Skills
12 4 t 1 4 21. 3t 1 10t 5t
3 32x 23. x 2 6 4 2
2 24. x 2 1 6x 1 8 4 x 1 4
26. 3x x1 9 4 (x + 3)
1 121 4 (k + 11) 27. 11k 7k 2 15
x 1x22
Mixed Review 59–76
(x 2 1)(x 2 2) 3
1 c2 2 1
1 19. s + 4 s 1 4
5(2x 2 5) x25
36. Answers may vary. Sample:
2 28. x2 1 10x 2 11 4 (x - 1)
x 1 12x 1 11
Error Prevention Exercises 7–28 When all factors of the numerator and denominator are eliminated, students may indicate a value of 0 instead of 1. Remind students that xx = 1.
Careers Exercises 38–40 A loan officer
31.
2 1 30. c2 1 3c 1 2 4 c 1 2 cc 1 c23 21 c 2 4c 1 3
7t2 2 28t ? 6t2 2 t 2 15 3t 2 5 7t 2 2t2 2 5t 2 12 49t3
2 2 22 33. x2 1 x 2 6 4 x2 1 5x 1 6 xx 2 3 x 2x26 x 1 4x 1 4
35. The student forgot to rewrite the expression using the reciprocal before canceling.
y24 42y 10 4 5 2 25. 2n2 2 5n 2 3 4 4n 1 5 4n 2 12n 2 7 2n 2 7
Multiply or divide. 2 2 29. t 1 5t 1 6 ? t2 2 2t 2 3 t ± 3 t23 t 1 3t 1 2
32.
Extension 49–53
Standardized Test Prep 54–58
22.
2x 1 4
Objective A B Core 16–28, 30–35, C
Divide. 20–28. See margin. 21 x13 20. x x14 4x14
Extension 47–48
37–40, 46
x22 15. A x 2 - 1 B ? 3x 13
4(t ± 1)(t ± 2) 3(2m ± 1)(m ± 2) Find the reciprocal of each expression. 2 x11 16. x 1 1 2
C
3. 5 2 ? 83 405
2. 3t ? 4t 122 t
m22 ? m 4. m 12 m21 1 2c 4c 7. 2c 1 2 ? cc 1 21c21
41–45
3(4x 1 1)
4. (m 1 2)(m 2 1)
Multiply.
Objective A B Core 1–15, 29, 36,
35. Error Analysis In the work shown at the right, what error did the student make in dividing the rational expressions? 36. Open-Ended Write two rational expressions. Find the product. 37. Critical Thinking For what values of x is the expression 2x 2 2 5x 2 12 6x
2 2 32. 5x 1 10x 2 215 4 2x2 1 7x 1 3 See 5 2 6x 1 x 4x 2 8x 2 5 left. 2 2 5 34. Q x2 2 25 RQ x2 1 x 2 20 R x 2 x
x 2 4x
x 1 10x 1 25
guides clients through the process of applying for a commercial, consumer, or mortgage loan. Loan officers must keep abreast of new types of loans and other financial products and services, so they can meet their customers’ needs. Enrichment 12-4
3a (a ⴙ 2)2 3a (a ⴙ 2)2 ⴜ ⴝ ⴜ aⴙ2 aⴚ4 aⴙ2 aⴚ4 aⴙ2 ⴝ 3a ⴜ aⴚ4 aⴚ4 ⴝ 3a ⴢ aⴙ2 3a(a ⴚ 4) ⴝ aⴙ2
2 12 4 23x undefined? 0, 4, –4 2 x 2 16
Reteaching 12-4 Practice 12-4 Name
Class
Multiplying and Dividing Rational Expressions
Find each product or quotient. 6 1. 59 ? 15
16 2. 83 4 27
16 3. a234 b 4 21
4. 29 4 a210 b 3
5. 18m2 4 9m 8 4m
6. 8x ? 4x 12 6
9 ? 25x 7. 15x 27
3 8. 12x 4 16x 25 5 2 11. 8n 4 20n 3 9
2 12. 14x 4 7x4 5
3 13. 4n ? 33n 11 36n2
3 14. 24r2 4 12r3 35r 14r
2 9 15. a 32 4 ? a 1 2
16. 4b 22 12 ? b 6b 23 5b
17. 2b ? 10 5 b2
18. b 2b 4 b 13 b13
19.
5y3 14y 7 ? 30y2
20.
4p 1 16 p14 4 5p 15p3
21.
3(h 1 2) 4h12 h13 h13
3 2 2 22. a 23 a ? a a2 1 a
2 6h ? 4h 1 12 23. hh 1 13 h16
2 2 1 n2 2 4 24. nn 1 ? 2 n11
2 26 25. x x2 x ? 3x 3x 2 3
2 10 ? 3 26. 5x x12 3x 2 6
2 16 4 3x 1 12 27. xx 2 x 24
x2 2 1 4 x 1 1 28. 3x 23 3
2 2 29. x 2 2 2x 2 24 ? x2 1 5x 1 6 x 2 5x 2 6 x 1 6x 1 8
2 2 30. x2 1 2x 2 35 ? x2 1 3x 2 18 x 1 4x 2 21 x 1 9x 1 18
2 2 31. 3x 1 14x 1 8 ? 2x 1 9x 2 5 2x2 1 7x 2 4 3x2 1 16x 1 5
2 2 32. 82 1 2x 2 x 4 x 22 11x 1 28 x 1 7x 1 10 x 2 x 2 42
2 2 2 x 2 6 ? 2x 29 33. x 3x 29 x 1 6x 1 9
2 2 x22 34. 6x 1213x 1 6 4 6x 1 4x 2 9 4x2 2 1
2 2 35. x 22 2x 2 35 4 x 12 7x 1 10 3x 1 27x 6x 1 12x
x2 2 x 2 6 4 x2 2 25 2x2 1 9x 1 10 2x2 1 15x 1 25
2 2 38. x2 2 4x 2 32 ? 3x2 1 17x 1 10 x 2 8x 2 48 3x 2 22x 2 16
659
6x3 4 9x2 9. 18x 10x4
4r3 ? 25 10. 10 16r2
36.
Lesson 12-4 Multiplying and Dividing Rational Expressions
Date
Practice 12-4
© Pearson Education, Inc. All rights reserved.
Check Understanding
14x 2 8x2 4 4x2 1 13x 2 12 37. 15 2 4x2 1 4x 2 15 3x2 1 13x 1 4
39.
2 9x2 2 16 4 6x2 1 11x 1 4 6x2 2 11x 1 4 8x 1 10x 1 3
40. Two darts are thrown at random onto the large rectangular region shown. Find the probability that both darts will land in the shaded region.
xⴙ1
2x ⴙ 2
3x
pages 659–661 Exercises
6x ⴙ 2
1 20. xx 2 1 3
23. –31
21. 6
24.
22. –12
25.
2(x 1 2) x21 n23 4n 1 5
26. 3x 4
Lesson 12-4 Practice
Algebra 1 Chapter 12
27. 7k 11 2 15 1 28. x 1 1
659
4. Assess
Loan Payments The formula below gives the monthly payment m on a loan when you know the amount borrowed A, the annual rate of interest r, and the number of months of the loan n. Use this formula and a calculator for Exercises 38–40. m5
Multiply or divide. 2 15 3x 1. 7x5 ? 14x 2
38. What is the monthly payment on a loan of $1500 at 8% annual interest for 18 months? $88.71
1 3 x2 1 9x 1 18 2. 6x x 1 6 ? 2x 1 1 3(x ± 3)
39. What is the monthly payment on a loan of $3000 at 6% annual interest for 24 months? $132.96
1 3 2 3. xx 1 1 (x + 5x + 6) 1 (x 1 1)(x 1 2)
40. Suppose your parents want to buy the house shown at the left. They have $15,000 for a down payment. Their mortgage will have an annual interest rate of 6%. The loan is to be repaid over a 30-year period. a. How much will your parents have to borrow? $100,000 b. How many monthly payments will there be? 360 payments c. What will the monthly payment be? $599.55 d. How much will it cost your parents to repay this mortgage over the 30-year period? $215,838
9x 2
1 8 4. 4x3x ? x 1 2 12x
5.
6.
2x 1 4 x2 1 11x 1 18 2(x 1 5) x 1 1 (x2 + 12x + x 1 9 x2 1 20x 1 99
r 11 r A Q 12 RQ 12 R n r Q 1 1 12 R 2 1
n
Lesson Quiz 12-4
x 1 1 x2 1 14x 1 45
11) ? x±1
2m 2 (m 1 2) (m 2 1)(m 1 4)
Geometry Find the volume of each rectangular solid. 41.
Alternative Assessment
2m 4 m
x5 3x 2
Number five sets of index cards each from 1 to 5 (25 cards total). Randomly distribute the cards to the class. Ask students to write their own rational expressions modeling the example whose number is on their card. Students should write their expressions directly on their cards. Collect the cards and redistribute. Have each student solve the expression on the card.
x22 4(x 1 7)
x2 x2 2x 35
3x 2 4
43.
m2 m 6 m2 m 2
Need Help? If two events A and B are independent, then P(A and B) = P(A) ? P(B).
660
m3 m2 m 12
44. 2a 1 2a2 7a 15
45. She wrote w 5 as a fraction so she could easily see what she could cancel.
660
42.
4a 6 3a 1 3a2 8a 3 2a2 5a 3
2 a 1 5
r2 9 r2 6r 9
r2 r2 1
r3 r2 3r 2 r 1 3 (r 2 1)(r 1 1) 2
45. Writing Robin’s first step in finding the product w2 ? w5 was to rewrite the 5 expression as w2 ? w1 . Why do you think Robin did this? See left. 46. Probability If a point is selected at random from a figure and is equally likely to be any point in the figure, then the probability that the point is in a shaded area of shaded part
part of the figure is area of whole figure . Suppose two points are chosen. x2 a. What is the probability that both 4(2x 1 1) 2 4x 4 points will be in the shaded part? b. What is the probability that one point will be in the shaded part x1 and the other point will not be in x(3x 1 2) the shaded part? 4(2x 1 1) 2 2x
Chapter 12 Rational Expressions and Functions
2x 1
Standardized Test Prep
47.
9m 2 (m 1 1) 2
50.
2(2a 1 3b)(a 1 2b) (5a 1 b)(2a 2 3b)
1 53. (w 1 2)(w 1 3)
a a 4 c .) Multiply or divide. (Hint: Remember that bc ≠ b d d 3 2 3m 2 2 2 2 48. 2 t 2 r 2 ? t 2 1 3tr 1 2r2 1 47. 3m ? (6m2 + 12m) 4m2 1 4m 2 8 t 1 tr 2 2r t 1 2tr 1 r 2 2 2 2 2 5xy 2 25x x 49. 2 5x 4 2 50. 2a2 2 ab 2 6b 2 4 2a 22 7ab 12 6b y 1 5 y 2 25 y 2 10y 1 25 2b 1 9ab 2 5a a 2 4b 3x w23 3m x(x 2 2) x2 2 1 w2 2 4 m22 51. m 22 1 2 2 9 2m(m 2 1) 52. 2(x 2 1) 53. 6 6m 2 m 2 2 ww 2 2 x 2x22
54. Simplify (2x - 5) ?
2x . 2x 2 9x 1 10 2
[1] one computational error OR answer with no work shown
Take It to the NET
25 C. 2x4x 15
D. 22x8x22510
3 G. a 2 a
F. -3
H.
a(a 2 3) a2
2 56. Which expression is equivalent to r r2 1 4 A 2r 2 - 2 B ? D
A.
r2
C.
r2
2 1 ? 1 (r 2 2 1) 2 r 21? r
I.
2r
a(a2 2 9) a2 (a 1 3)
F. Multiply the numerators. H. Factor each polynomial. 58. Simplify
?
75.
x5
y
16 G. Find the reciprocal of 2x 2x 1 2 .
I. Multiply the denominators.
O
2
8
Write the product in factored form. Show your work.
24
3 3 60. 36k4 4k 48k
2 14 7 61. 7m 3m 2 6 3
7q 5 q4 28q 4
2 2 63. 15t 242 27 5t 82 9
64.
5a 2 1 2 10a 4 2 15a 2 2a 2 3
66.
2z2 2 11z 2 21 2z 1 3 z2 2 6z 2 7 z 1 1
69. a = 3.1, b = 4.3 5.3
1 0.2 71. a = !10, b = !111 11 72. a = 15, b = 12
16
y
12
6m3 m2 2 2 2 3m 12m 2 18m 2 2c 2 9 67. 4c 2 2 36c 1 81 2c 1 8 4c 2 2c 2 72
Assume a and b are legs of a right triangle, and c is the hypotenuse. Find the length of the missing side of each right triangle. If necessary, round to the nearest tenth. 68. a = 2, b = 8 8.2
(5, 27)
76.
Simplify each expression.
65.
x
8
16
2 25 b 2 5 59. 5b 10 2
Lesson 10-2
24
(5, 27)
Mixed Review
62.
2
16
See left.
Lesson 11-2
O
r2
2 2x 2 3 1 6 57. Which CANNOT be the first step in multiplying x 2 by 2x x 1 3 2x 1 2 ? G
3x x 2 1.
8
2 21? r2 2 1 r 2 D. r 2 1 ? 21 r 2r 2 2
B.
Q 12 2 2 R
x2 2 1 x
y x
a
a
Web Code: aea-1204
Lesson 12-3
x 5
74.
2 a 1 3 55. Simplify a 22 9 4 . G
Online lesson quiz at www.PHSchool.com
Short Response
pages 659–661 Exercises B
B. x 2x 22
A. 1 2 58. [2] x x2 1 ? x 3x 21 ≠ (x 1 1)(x 2 1) (3x) ≠ x ( x 2 1) 3(x ± 1)
Exercise 56 Remind students to multiply by the reciprocal of 2r 2 - 2.
Standardized Test Prep Standardized Test Prep Multiple Choice
Resources For additional practice with a variety of test item formats: • Standardized Test Prep, pp. 697–699 • Test-Taking Strategies, p. 692 • Test-Taking Strategies with Transparencies
8
(
41 , 487
)
x 41 x
2
O
2
70. a = !7, c = !32 5 73. a = 2 13, b = 6 23 7.1
Graph each function. Label the axis of symmetry and the vertex. 74–76. See margin. 74. y = x 2 + 10x - 2
75. y = x 2 - 10x - 2
76. y = 2x 2 + x + 5
Lesson 12-4 Multiplying and Dividing Rational Expressions
661
661
