rational expressions a rational expression is any expression that consists of a polynomial divided...
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Rational Expressions
A rational expression is any expression that consists of a polynomial divided by a nonzero polynomial.
Examples of rational expressions:8
,11
6,
5x
2
2
3 8,
8 9
x x
x
13 8y
y
Domain: ,0 0,x
Domain: { 0}x x
2( )
3
xf x
x
Domain is ,3 3,x
Zero is 2x
Vertical Asymptote is 3x
Example: Consider the function, determine the following.
b) The domain of f (x).
a) Graph f (x) in the window:
2
2
6 7( )
2 15
x xf x
x x
8,10 7,7
2
2
6 7( )
2 15
x xf x
x x
( 7)( 1)
( 5)( 3)
x x
x x
So, 3 or 5 , , 3 3,5 5,x x x x
c) Find the zeros/roots for f (x).
d) Find the vertical asymptotes.
( 7)( 1)( ) 0, so let ( 7)( 1) 0
( 5)( 3)
x xf x x x
x x
The zeros are 1 or 7.x x
5x 3x
( 5)( 3) 0, so let 3 and 5 are vertical asymptotes.x x x x
Example
Simplify:
Solution
29
36
x
x
29
936
9
4
x
x
x x
x
9
94
x x
x
14
x
4
x
Factoring the numerator and the denominator. Note the common factor of 9x.
Rewriting as a product of two rational expressions.
Removing the factor 1
9x/9x = 1, clever form of 1
Example
Simplify
a) b)2
2
8 28
4 12
a a
a a
2
2
2 15
9
x x
x
2
2
4
4
8 28 (2 7)
4 12 ( 3)
a
a
a a a
a a a
24
34
7a
a
a
a
71
2
3
a
a
2 7
3
a
a
2
2
( 32 15 () 5)
9 (3 3( ))
x x x
x
x
xx
5
3 3
3 x
x
x
x
51
3
x
x
5
3
x
x
Example Multiply. Then simplify by removing a factor of 1. a) b)
Solutiona)
46 5
10 6
a
a
2
2
5 4 3
9 4
x x x
x x
4 46 (5)
10(
6 5
6 610 )
aa
a a
2 3 5
2 5 2 3
a a a a
a
2
3 a 5a a a 2 5 2 3 a
3
2
a
Example continued
b)
2 2
2 2
5 4 3 ( 5 4)( 3)
9 4 ( 9)( 4)
x x x x x x
x x x x
2
2
5 4 3
9 4
x x x
x x
( 3)( 4)( 1)
()( 4)( 3 3)
x x
x xx
x
( 4)x
( 1) ( 3)x x
( 3) ( 3)x x ( 4)x
1
3
x
x
Solution
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example Simplify.
NO “C” BOMBS!! Look for clever forms of one!!
Don’t let the “TERM-ITE” bite you!!
Example
Divide and, if possible, simplify:
Solution
2 2
2 2.
4 4 4
x x
x x x
2
2 2 2
2 2 2 4 4
4 4 4 4 2
x x x x x
x x x x x
( 2)( 2)
(
( 2)
( 22 ) ))( 2
x x
x
x
x x
( 2)x
( 2)x ( 2)
( 2)
x
x
( 2)x ( 2)x
( 2)
( 2)
x
x
Example
Divide and, if possible, simplify:
Solution
2 2
2 2
3 2 5 6.
5 4 10 24
x x x x
x x x x
( 1)( 2)( 6)( 4)
( 4)( 1)( 2)( 3)
x x x x
x x x x
2 2 2 2
2 2 2 2
3 2 5 6 3 2 10 24
5 4 10 24 5 4 5 6
x x x x x x x x
x x x x x x x x
( 1)x
( 2)x ( 6) ( 4)x x
( 4)x ( 1)x ( 2)x ( 3)x
( 6)
( 3)
x
x
