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Lesson 7.2 Exercises, pages 537–543
A
3. Simplify each expression.
3 2b
a) 4 5
#
3
4
3b
ⴝ
10
ⴝ
p2 15
p2
ⴝ
p
5
5
#
ⴝ
c3
#
ⴚ2 4 x
3
ⴝ ⴚ6x
2
d) 10
15 3
p
#
3
9x
2
# -2c5
Non-permissible value: c ⴝ 0
c3
10
# ⴚ5 ⴝ
2c
ⴝ
6
#
ⴝ 3p, p ⴝ 0
- 4x 9x
2
c) 3
#
Non-permissible value: p ⴝ 0
2b
5
#
2
p2 15
p
b) 5
2
c3
2 10
# ⴚ5
2c
ⴚc
,cⴝ 0
4
2
7.2 Multiplying and Dividing Rational Expressions—Solutions
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4. Simplify each expression.
4
2
a) n , n
Quit
2b
4
b) 5 ,
10b
Non-permissible value: n ⴝ 0
Non-permissible value: b ⴝ 0
2
4
2
n ⴜ nⴝ n
2b
4
2b
ⴜ
ⴝ
5
5
10b
#
n
42
1
2
ⴝ ,nⴝ 0
2
#
10 b
42
ⴝ b2, b ⴝ 0
d3
6
c) x , 3
d
d) 4 , 12
Non-permissible value: x ⴝ 0
2
6
6
x ⴜ 3ⴝ x
#
Non-permissible value: d ⴝ 0
d3
d
d3
ⴜ
ⴝ
4
12
4
1
3
2
5. Simplify each expression.
2
m 10p
a) 5p 3m
#
Non-permissible values:
p ⴝ 0 and m ⴝ 0
#
ⴝ
2
#
#
12 3
d
ⴝ 3d2, d ⴝ 0
ⴝ x, x ⴝ 0
2
m 10p
m
ⴝ
5p 3m
5 p
2
10 p 2
3m
b)
- 2ab2
- 4a
,
15ab
5a3b
Non-permissible values:
a ⴝ 0 and b ⴝ 0
ⴚ 2ab2
ⴚ 4a
ⴚ 2 a b2
ⴜ
ⴝ
2
3
15ab
5a b
5 a3 b
2p
, p ⴝ 0, m ⴝ 0
3
ⴝ
#
3
15 a b
ⴚ4 a
2
3b2
, a ⴝ 0, b ⴝ 0
2a2
B
6. Simplify each expression.
8(x - 3)
3x
a)
2(x - 3)
9x2
#
Non-permissible values:
x ⴝ 0 and x ⴝ 3
©P
4
ⴝ
3 x
2 (x ⴚ 3)
ⴝ
4
, x ⴝ 0, 3
3x
DO NOT COPY.
#
8 (x ⴚ 3)
3 9 x
2
b)
15(x + 5)
#
2x2
8x
5(x + 5)2
Non-permissible values:
x ⴝ 0 and x ⴝ ⴚ 5
3
ⴝ
ⴝ
15 (x ⴙ 5)
2x
2
#
4
8 x
5 (x ⴙ 5) 2
12
, x ⴝ ⴚ5, 0
x(x ⴙ 5)
7.2 Multiplying and Dividing Rational Expressions—Solutions
7
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c)
2(x - 4)(x + 5)
4(x + 5)
,
3(x + 1)
(x + 1)2
d)
2 (x ⴚ 4) (x ⴙ 5)
#
3 (x ⴙ 1)
(x ⴙ 1) 2
(x ⴚ 4)(x ⴙ 1)
ⴝ
, x ⴝ ⴚ5, ⴚ1
6
#
,
3
(y - 3)
ⴝ
2 4 (x ⴙ 5)
7. Simplify each expression.
2a + 2b a - 2b
a)
3a - 6b a + b
y(y + 1)
10y
(y - 3)2
Non-permissible values:
y ⴝ 3, y ⴝ 0, and y ⴝ ⴚ1
Non-permissible values:
x ⴝ ⴚ5 and x ⴝ ⴚ1
ⴝ
Quit
ⴝ
b)
10 y
#
(y ⴚ 3)
3
(y ⴚ 3)2
y (y ⴙ 1)
10
, y ⴝ ⴚ1, 0, 3
(y ⴚ 3)(y ⴙ 1)
n2 - 4 6n + 3
2n + 1 n - 2
#
Non-permissible values:
Non-permissible values:
a ⴝ 2b and a ⴝ ⴚb
n ⴝ ⴚ and n ⴝ 2
2a ⴙ 2b a ⴚ 2b
3a ⴚ 6b a ⴙ b
n2 ⴚ 4 6n ⴙ 3
2n ⴙ 1 n ⴚ 2
1
2
#
ⴝ
2 (a ⴙ b)
#
3 (a ⴚ 2b)
#
a ⴚ 2b
aⴙb
2
3
ⴝ 3(n ⴙ 2), n ⴝ ⴚ , 2
x2 - 25
x + 5
c) x - 4 , 3x - 12
d)
(x ⴚ 5)(x ⴙ 5)
xⴙ5
ⴜ
xⴚ4
3(x ⴚ 4)
(b + 2)2
3b2 - 3
ⴝ
Non-permissible values:
ⴝ
x ⴝ ⴚ5, x ⴝ 4, and x ⴝ 5
3 (x ⴚ 4)
ⴝ
xⴙ5
xⴚ4
ⴝ
3
, x ⴝ ⴚ5, 4, 5
xⴚ5
#
#
1
2
ⴝ , a ⴝ 2b, ⴚb
ⴝ
(n ⴚ 2) (n ⴙ 2) 3 (2n ⴙ 1)
2n ⴙ 1
nⴚ2
ⴝ
(b ⴙ 2)2
3(b ⴚ 1)
2
ⴝ
(x ⴚ 3)(x ⴙ 2) (x ⴚ 4)(x ⴙ 4)
xⴙ4
x(x ⴙ 2)
#
Non-permissible values:
x ⴝ ⴚ4, ⴚ2, 0
(x ⴚ 3) (x ⴙ 2)
ⴝ
xⴙ4
#
(x ⴚ 4) (x ⴙ 4)
x (x ⴙ 2)
(x ⴚ 3)(x ⴚ 4)
, x ⴝ ⴚ4, ⴚ 2, 0
ⴝ
x
ⴜ
2(b ⴙ 2)
1ⴚb
(b ⴙ 2)2
3(b ⴚ 1)(b ⴙ 1)
ⴝ
2(b ⴙ 2)
ⴜ
(b ⴙ 2) 2
3 (b ⴚ 1) (b ⴙ 1)
ⴝⴚ
#
2b + 4
1 - b
ⴚ (b ⴚ 1)
Non-permissible values:
b ⴝ ⴚ2, b ⴝ ⴚ1, and b ⴝ 1
(x ⴚ 5) (x ⴙ 5)
8. Simplify each expression.
x2 - x - 6 x2 - 16
a)
x + 4
x2 + 2x
,
b)
# ⴚ (b ⴚ 1)
2 (b ⴙ 2)
bⴙ2
, b ⴝ ⴚ2, ⴚ1, 1
6(b ⴙ 1)
2x2 - x - 1 4x2 + 28x + 48
x2 + 2x - 3 2x2 - 13x - 7
#
ⴝ
ⴝ
(2x ⴙ 1)(x ⴚ 1) 4(x2 ⴙ 7x ⴙ 12)
(x ⴙ 3)(x ⴚ 1)
#
(2x ⴙ 1)(x ⴚ 7)
(2x ⴙ 1)(x ⴚ 1) 4(x ⴙ 3)(x ⴙ 4)
(x ⴙ 3)(x ⴚ 1)
#
(2x ⴙ 1)(x ⴚ 7)
Non-permissible values:
1
2
x ⴝ ⴚ3, ⴚ , 1, 7
ⴝ
(2x ⴙ 1) (x ⴚ 1)
(x ⴙ 3) (x ⴚ 1)
# 4 (x ⴙ 3) (x ⴙ 4)
(2x ⴙ 1) (x ⴚ 7)
4(x ⴙ 4)
1
, x ⴝ ⴚ3, ⴚ , 1, 7
ⴝ
xⴚ7
2
8
7.2 Multiplying and Dividing Rational Expressions—Solutions
DO NOT COPY.
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c)
3m - 12
m2 - m - 12
, 2
m + 3
m - 9
ⴝ
(m ⴚ 4)(m ⴙ 3)
3(m ⴚ 4)
ⴜ
mⴙ3
(m ⴚ 3)(m ⴙ 3)
d)
4s2 + 4s + 1
2s2 - 7s - 4
,
2
12s + 8
6s - 5s - 6
ⴝ
Non-permissible values:
m ⴝ ⴚ3, 3, 4
Quit
(2s ⴙ 1)(s ⴚ 4)
(3s ⴙ 2)(2s ⴚ 3)
ⴜ
(2s ⴙ 1)(2s ⴙ 1)
4(3s ⴙ 2)
Non-permissible values:
ⴝ
(m ⴚ 4) (m ⴙ 3)
mⴙ3
ⴝ
(m ⴚ 3)(m ⴙ 3)
, m ⴝ ⴚ3, 3, 4
3
2 1 3
3 2 2
(2s ⴙ 1) (s ⴚ 4)
s ⴝⴚ ,ⴚ ,
# (m ⴚ 3)(m ⴙ 3)
ⴝ
3 (m ⴚ 4)
ⴝ
#
4 (3s ⴙ 2)
(3s ⴙ 2) (2s ⴚ 3) (2s ⴙ 1) (2s ⴙ 1)
4(s ⴚ 4)
(2s ⴚ 3)(2s ⴙ 1)
2
3
1 3
2 2
,sⴝⴚ , ⴚ ,
9. Each rational expression is the product of two different rational
expressions. What might the rational expressions have been?
Describe the strategy you used to find out.
3
a) y
b)
x + 1
3x
I wrote two polynomial expressions such that when I divided the
numerators and denominators by their common factors, I was left with
the given product. I then expanded where necessary.
Sample Response:
(x ⴙ 1)(x ⴙ 2)
(2)
xⴙ1
ⴝ
3x
3x(2)
(x ⴙ 2)
3(2x) (5)
3
y ⴝ (5) (2x)y
#
ⴝ
#
6x 5
, x ⴝ 0, y ⴝ 0
5 2xy
ⴝ
#
x2 ⴙ 3x ⴙ 2
2
, x ⴝ ⴚ2, 0
6x
xⴙ2
#
10. Each rational expression is the quotient of two different rational
expressions. What might the rational expressions have been?
How do you know?
3
a) x - 5
b)
x - 9
x + 8
I wrote a product of two polynomial expressions such that when I divided
the numerators and denominators by their common factors, I was left with
the given expression. I then expanded where necessary, inverted the
second expression, then changed multiplication to division.
Sample Response:
3(5)
3
xⴙ4
ⴝ
xⴚ5
(x ⴚ 5)(x ⴙ 4)
(5)
#
ⴝ
15
5
ⴜ
,
xⴙ4
(x ⴚ 5)(x ⴙ 4)
x ⴝ ⴚ4, 5
©P
DO NOT COPY.
(x ⴚ 9)2
xⴚ9
2
ⴝ
xⴙ8
2(x ⴙ 8) (x ⴚ 9)
#
ⴝ
x2 ⴚ 18x ⴙ 81
xⴚ9
ⴜ
,
2x ⴙ 16
2
x ⴝ ⴚ 8, 9
7.2 Multiplying and Dividing Rational Expressions—Solutions
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11. Simplify each expression.
3x3y 2y2 8x3y2
a)
18x3 4xy5 12xy
#
#
3 x3 y
36
ⴝ
#
4 x y5
18 x 3
3
#
#
Non-permissible values:
a ⴝ 0, b ⴝ 0
2
2
2 y2
3ab
8b2
2a2b2
,
4ab3 27a5b3
6ab3
b)
Non-permissible values:
x ⴝ 0, y ⴝ 0
ⴝ
Quit
8 x 3 y2
6 12 x y
3 a b
4 a b3
ⴝ
2
#
3
8 b2
5 32
9 27 a b
#
2
6 a b3
2 a 2 b2
2
ⴝ 6 2 , a ⴝ 0, b ⴝ 0
3a b
x
, x ⴝ 0, y ⴝ 0
18y
12. Simplify each expression.
2x - 10
2x - 4
x + 3
a) 2
, 2
x - 2x - 15 2x2 + 15x + 7
x + 5x - 14
#
ⴝ
2(x ⴚ 5)
2(x ⴚ 2)
xⴙ3
ⴜ
(x ⴚ 5)(x ⴙ 3) (2x ⴙ 1)(x ⴙ 7)
(x ⴙ 7)(x ⴚ 2)
#
1
2
(x ⴙ 7) (x ⴚ 2)
Non-permissible values: x ⴝ ⴚ7, ⴚ3, ⴚ , 2, 5
b)
2 (x ⴚ 5)
ⴝ
xⴙ3
(x ⴚ 5) (x ⴙ 3)
ⴝ
1
1
, x ⴝ ⴚ7, ⴚ3, ⴚ , 2, 5
2x ⴙ 1
2
#
#
(2x ⴙ 1) (x ⴙ 7)
2 (x ⴚ 2)
x2 + 14x + 49
3x + 24
x2 - x - 56
, 2
x + 8
x - 6x - 16 x2 - 16x + 64
#
ⴝ
(x ⴚ 8)(x ⴙ 7)
(x ⴙ 7)(x ⴙ 7)
3(x ⴙ 8)
ⴜ
xⴙ8
(x ⴚ 8)(x ⴙ 2) (x ⴚ 8)(x ⴚ 8)
#
Non-permissible values: x ⴝ ⴚ8, ⴚ7, ⴚ2, 8
(x ⴚ 8) (x ⴙ 7) (x ⴚ 8) (x ⴙ 2)
xⴙ8
(x ⴙ 7) (x ⴙ 7)
3(x ⴙ 2)
ⴝ
, x ⴝ ⴚ8, ⴚ7, ⴚ2, 8
xⴙ7
ⴝ
#
x + 4y x2 - 25y2
#
3 (x ⴙ 8)
(x ⴚ 8) (x ⴚ 8)
x + 5y
, x - 4y
c) x - 5y # 2
x - 16y2
ⴝ
x ⴙ 4y (x ⴚ 5y)(x ⴙ 5y)
x ⴙ 5y
ⴜ
x ⴚ 5y (x ⴚ 4y)(x ⴙ 4y)
x ⴚ 4y
#
Non-permissible values: x ⴝ ⴚ 5y, ⴚ4y, 4y, 5y
ⴝ
x ⴙ 4y
x ⴚ 5y
#
(x ⴚ 5y) (x ⴙ 5y)
(x ⴚ 4y) (x ⴙ 4y)
#
x ⴚ 4y
x ⴙ 5y
ⴝ 1, x ⴝ ⴚ5y, ⴚ4y, 4y, 5y
10
7.2 Multiplying and Dividing Rational Expressions—Solutions
DO NOT COPY.
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d)
Quit
(5c + 6d)2 9c2 - 16d2
3c + 4d
,
3c - 4d 25c2 - 36d2
5c - 6d
#
ⴝ
(5c ⴙ 6d)2 (3c ⴚ 4d)(3c ⴙ 4d)
3c ⴚ 4d
#
(5c ⴚ 6d)(5c ⴙ 6d)
Non-permissible values: c ⴝ ⴚ
ⴝ
(5c ⴙ 6d) 2
3c ⴚ 4d
#
4d
6d 6d 4d
,ⴚ , ,
3
5 5 3
(3c ⴚ 4d) (3c ⴙ 4d)
(5c ⴚ 6d) (5c ⴙ 6d)
ⴝ 5c ⴙ 6d, c ⴝ ⴚ
3c ⴙ 4d
5c ⴚ 6d
ⴜ
5c ⴚ 6d
3c ⴙ 4d
#
4d
6d 6d 4d
,ⴚ , ,
3
5 5 3
13. Simplify each expression.
x2 - x - 6
x2 - 2x - 8
, 2
x + 5
x + 2x - 15
2
x - x - 6
x2 - 2x - 8
,
2
x + 5
x + 2x - 15
Compare the solutions.
What do you notice?
x2 ⴚ x ⴚ 6
x2 ⴚ 2x ⴚ 8
ⴜ 2
xⴙ5
x ⴙ 2x ⴚ 15
ⴝ
(x ⴚ 4)(x ⴙ 2)
(x ⴚ 3)(x ⴙ 2)
ⴜ
xⴙ5
(x ⴙ 5)(x ⴚ 3)
Non-permissible values:
x ⴝ ⴚ5, ⴚ2, 3
ⴝ
(x ⴚ 4) (x ⴙ 2)
xⴙ5
#
x2 ⴚ x ⴚ 6
x2 ⴚ 2x ⴚ 8
ⴜ
xⴙ5
x ⴙ 2x ⴚ 15
2
ⴝ
(x ⴚ 3)(x ⴙ 2)
(x ⴙ 5)(x ⴚ 3)
(x ⴚ 4)(x ⴙ 2)
xⴙ5
ⴜ
Non-permissible values:
x ⴝ ⴚ5, ⴚ2, 3, 4
(x ⴙ 5) (x ⴚ 3)
(x ⴚ 3) (x ⴙ 2)
ⴝ x ⴚ 4, x ⴝ ⴚ 5, ⴚ2, 3
The solutions are reciprocals.
(x ⴚ 3) (x ⴙ 2)
xⴙ5
(x ⴙ 5) (x ⴚ 3) (x ⴚ 4) (x ⴙ 2)
1
, x ⴝ ⴚ5, ⴚ2, 3, 4
ⴝ
xⴚ4
ⴝ
#
C
14. a) Write a polynomial A so that the expression below simplifies to -1.
3n2 + 2n - 8
A
4
, n Z -2, - 1, 3
n2 + 4n + 4 3n2 - n - 4
#
ⴝ
ⴝ
(3n ⴚ 4) (n ⴙ 2)
(n ⴙ 2) (n ⴙ 2)
#
A
(3n ⴚ 4) (n ⴙ 1)
A
(n ⴙ 2)(n ⴙ 1)
Set the expression equal to ⴚ1.
ⴚ1 ⴝ
A
(n ⴙ 2)(n ⴙ 1)
A ⴝ ⴚ (n ⴙ 2)(n ⴙ 1)
A ⴝ ⴚ(n2 ⴙ 3n ⴙ 2)
A ⴝ ⴚ n2 ⴚ 3n ⴚ 2
©P
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7.2 Multiplying and Dividing Rational Expressions—Solutions
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1
b) Write a polynomial B so that the expression below simplifies to 5 .
x2 + 6x + 9
3x2 + 11x + 6
,
, x Z -3, 0
2
B
5x + 15x
ⴝ
ⴝ
ⴝ
(x ⴙ 3)(x ⴙ 3)
5x(x ⴙ 3)
(x ⴙ 3) (x ⴙ 3)
5x (x ⴙ 3)
(x ⴙ 3)(3x ⴙ 2)
B
ⴜ
#
B
(x ⴙ 3) (3x ⴙ 2)
B
5x(3x ⴙ 2)
1
5
Set the expression equal to .
1
B
ⴝ
5
5x(3x ⴙ 2)
Bⴝ
5x(3x ⴙ 2)
5
B ⴝ x(3x ⴙ 2)
B ⴝ 3x2 ⴙ 2x
2
Since B is in the denominator, x ⴝ ⴚ , 0
3
2
1 - a
15. Simplify:
1 -
4
a2
aⴚ2
a
ⴝ 2
a ⴚ4
a2
aⴚ2
a2
2
a
a ⴚ4
aⴚ2
a2
ⴝ
a
(a ⴚ 2) (a ⴙ 2)
ⴝ
#
#
ⴝ
12
Non-permissible values: a ⴝ ⴚ2, 0, 2
a
, a ⴝ ⴚ2, 0, 2
aⴙ2
7.2 Multiplying and Dividing Rational Expressions—Solutions
DO NOT COPY.
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