12-2 Simplify Rational Expressions
Name
Date
Simplify x 2 8x 12 . Identify the excluded values.
x 4x 4
(x 6)(x 2)
Factor the numerator and
(x 2)(x 2)
denominator.
2
1
( x 6 )( x 2 )
x6
( x 2 )( x 2 )
x2
So in simplest form,
x20
Divide out common
factors.
1
x2 8x 12
x2 4x 4
2.
The excluded value is 2.
1
1
1
5 b (1 2 b)
5b
;
(2 b 1)(3 b 2) 3b 2
1
1
y 2 0; y 4 0
y2
; y fi 4, y fi 2
y4
5.
x2
2b 1 0; 3b 2 0
2
1
5b
;bfi ,bfi
3
2
3b 2
x2 16
10x 24
6.
x2
x2 9
9x 18
1
1
1
8 c (1 3 c )
8c
;
(3 c 1)(3 c 4 ) 3c 4
( x 4 )( x 4 ) x 4
;
( x 4 ) ( x 6) x 6
( x 3)( x 3) x 3
;
( x 3)( x 6) x 6
3c 1 0; 3c 4 0
1
8c
4
;cfi ,cfi
3
3c 4
3
x 4 0; x 6 0
x4
; x fi 4, x fi 6
x6
x 3 0; x 6 0
x3
; x fi 3, x fi 6
x6
1
1
Copyright © by William H. Sadlier, Inc. All rights reserved.
10b2
3. 5b
6b2 7b 2
( y 2)( y 2) y 2
;
( y 2)( y 4) y 4
1
x1
;
x
1
0,
x
30
x3
x 1 ; x ⬆ 3, x ⬆ 1
x3
8c 24c2
9c2 15c 4
Simplify.
y2 4
y2 6y 8
( x 1)( x 1)
(x 1)(x 1)
(x 1)(x 3) ( x 1)( x 3)
4.
Apply the Zero-Product Property.
x 2
is x 6.
x2
Simplify. Identify excluded values.
2
1. 2 x 1
x 4x 3
Excluded values:
(x 2)(x 2)
Factor the denominator.
2
7. 4x2 4x 3
4x 8x 5
1
2
8. 18m2 60m 18
9m 21m 6
4
3
9. w8 w6
w w
1
(2 x 3) (2 x 1) 2x 3
;
( 2 x 1) ( 2 x 5) 2x 5
6 ( m 3) (3 m 1) 2(m 3)
;
m2
3 (3 m 1)( m 2)
2x 1 0; 2x 5 0
5
1
2x 3
;xfi ,xfi
2
2
2x 5
3m 1 0; m 2 0
1
2(m 3)
; m fi 2, m fi 3
m2
1
10.
q7 q6
q9 q7
1
1
1
; w6 0; w 1 0; w 1 0
w3(w 1)
1
; w fi 1, w fi 0, w fi 1
w3(w 1)
1
2
11. 9y2 9y 10
9y 12y 5
2
12. 8a 2 4a 24
16a 8a 24
1
1
q6 ( q 1)
(q 1)
;
q7 (q 1)(q 1) q7 ( q 1)( q 1)
q6
w 3 (w 1)
w3(w 1)
;
w6(w2 1) w 6 (w 1)(w 1)
1
2
1
1
1
; q fi 0; q fi 1; q fi 1
q (q 1)
(3y 2)(3y 5)
;
(3y 1)(3y 5)
( 3 y 2) ( 3 y 5)
(3 y 1) (3 y 5)
3y 2
5
1
;yfi ;yfi
3
3
3y 1
Lesson 12-2, pages 308–309.
1
1
1
4(2a 3)(a 2) 4 (2 a 3)( a 2)
;
8(a 1)(2a 3) 8 ( a 1) (2 a 3)
2
a2
3
;afi ;afi1
2
2(a 1)
1
Chapter 12 305
For More Practice Go To:
Simplify each expression. Identify excluded values.
y 4
1
14. 2
13. m
m2 1
y
16
1
1
(
)
m 1
( y 4 )
1
;
(m 1) (m 1)
y4
y
4
y
4
(
)(
)
1
2
16. b2 3b 1
b 7b 8
3
2
17. 3n3 14n2 5n
2n 11n 5n
1
(3z 2)(z 3) ( 3 z 2 )( z 3 )
;
(3z 2)(z 2) ( 3 z 2 )( z 1)
1
z3
; 3z 1 2 5 0; z 1 5 0
z1
2
z3
;zfi ;zfi1
3
z1
m(4 m 1)( m 2)
m ( m 2) ( 3 m 2)
1
n 5 0; 2n 1 0; n 0
1
3n 1
; n fi 5, n fi ; n fi 0
2
2n 1
2
20. c2 c 2
c 3c 2
1
(c 1)(c 2) ( c 1)( c 2 )
;
(c 1) (c 2) ( c 1)( c 2 )
1
c2
; c 1 1 5 0; c 1 2 5 0
c2
c2
; c fi 1; c fi 2
c2
22. The length of Tom’s business card is 2 cm more
than the width. Anne’s business card is 2 cm
wider and 4 cm longer than Tom’s business
card. What is the ratio of the area of Tom’s
business card to Anne’s business card?
Let w width of Tom’s card
length w 2, area of Tom’s w(w 2)
width of Anne’s w 2, length w 6
area (w 2)(w 6);
w
w(w 2)
ratio of areas: (
w 2)(w 6) w 6
1
1
n (3 n 1)(n 5)
n (n 5) (2 n 1)
1
r2 r 5
in simplest form
r2 3r 4
(r 1)(r 4); r 1 0; r 4 0
in simplest form; r 1, r 4
3
2
18. 4m3 7m2 2m
3m 4m 4m
1
1
b2 3b 1
in simplest form
b2 7b 8
(b 1)(b 8); b 1 0; b 8 0
in simplest form; b 1, b 8
2
19. 3z 2 7z 6
3z z 2
1
y 4 0; y 4 0
1
y 4 ; y fi 4, y fi 4
1
1
m 0; m 2 0; 3m 2 0
2
4m 1
; m , m 0, m 2
3
3m 2
2
21. 2d2 5d 3
4d 12d 5
1
( 2 d 1)( d 3 )
(2d 1)(d 3)
;
(2d 1) (2d 5) ( 2 d 1)( 2 d 5 )
1
d3
; 2d 1 1 5 0; 2d 5 5 0
2d 5
5
1
d3
;dfi ;dfi
2
2
2d 5
23. Let s represent the length of an edge of a cube.
If the lengths of the edges are all increased by
3, what will the ratio of surface area to volume
of the new cube be? If the original length was
10 cm, what will the ratio of surface area to
volume be? (Hint: SA 6s2)
SA 6(s 3)2 and V (s 3)3,
6
SA 6(s 3)2
(s 3)3
s3
V
6
6
for s 10,
s 3 13
24. The ratio of the area of a square to the area of a right triangle is 1. The height of the right triangle
is 3 times the length of the side of the square. What is the ratio of the length of the side of the
square to the base of the triangle?
1
Area of a square s2 and area of a right triangle bh; ratio of the area of the square to the area of
2
2
2s2
2s2
2s
2•3
2s2
s
the triangle:
1;
1; h 3s, so
to get
1. Since
1,
1. Simplify
3b
3•2
bh
3bs
3bs
1 bh
2
2s
s
3
and
1, then .
3b
b 2
306 Chapter 12
Copyright © by William H. Sadlier, Inc. All rights reserved.
1
m 1; m 1 1 0; m 2 1 0
1
m 1; m fi 1, m fi 1
2
15. r2 r 5
r 3r 4