Simplify each rational expression, if possible.
r+2
b 2 - 25
3a. __
3b. __
2
2
r + 7r + 10
b + 10b + 25
Recall from Chapter 8 that opposite binomials can help you factor polynomials.
Recognizing opposite binomials can also help you simplify rational expressions.
x-3
Consider ____
. The numerator and denominator are opposite binomials.
3-x
Therefore,
x-3 =_
x-3 =_
x - 31 = _
1 = -1.
_
3-x
-x + 3
-1(x - 3) -1
EXAMPLE
4
Simplifying Rational Expressions Using Opposite Binomials
Simplify each rational expression, if possible.
A
2x - 10
_
B
25 - x 2
2(x - 5)
__
Factor.
2 - 2m
__
2m 2 + 2m - 4
2(1 - m)
__
(5 - x)(5 + x)
2(m + 2)(m - 1)
2(x - 5)
2(1 - m)
__
Identify opposite binomials. __
(5 - x)(5 + x)
2(m + 2)(m - 1)
2(x - 5)
2(1 - m)
__
___
Rewrite one opposite
-1(x - 5)(5 + x)
2(m + 2)(-1)(1 - m)
binomial.
2(x - 5)
2(1 - m)1
__
Divide out common factors. __
-1(x - 5)(5 + x)
2(m + 2)(-1)(1 - m)
2
-_
5+x
1
-_
m+2
Simplify.
Simplify each rational expression, if possible.
3x - 12
4a. _
16 - x 2
6 - 2x
4b. __
2
2x - 4x - 6
3x - 33 tude
4c. _
x 2 - 121
Opposite Binomials
I didn’t understand why the quotient of opposite binomials simplified to –1.
My teacher showed me an example on a number line:
Î
£ä
The distance between 3 and 10 is always the same (7 units). But depending on
the order of the subtraction, the difference could be positive or negative.
Tanika Brown,
Washington High
School
644
10 - 3 = 7
3 - 10 = -7
So whenever you divide something in the form
by its opposite, which is always -1.
Chapter 10 Rational Functions and Equations
a-b
_____
, you get a number divided
b-a