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Simplifying, Multiplying and Dividing Rational Expressions
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Simplifying Rational Expressions Let a, b, and c be nonzero real
numbers (or variable expressions). Then,
cb
ca
b
a
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Example: Simplify
Factor the numerator and denominator
Cancel common factors Rewrite the simplified expression
1
652
2
x
xx
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Example: Simplify1
652
2
x
xx
1
652
2
x
xx
)1)(1(
)1)(6(
xx
xx
1
6
x
x
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Multiplying Rational Expressions with Monomials Simplify as much as possible Multiply numerators Multiply denominators Simplify Rewrite the simplified expression
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Example: 2
43
22
32
18
10
2
6
y
yx
yx
yx
2
43
22
32
18
10
2
6
y
yx
yx
yx
36
60 33yx
3
5 33yx
2 5
3
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Multiplying Rational Expressions with polynomials Factor the numerators and
denominators of each rational expression
Simplify by dividing out common factors
Multiply Rewrite in simplified form (no
parentheses or negative exponents)
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Example:x
xx
xx
xx
3
143
123
273 2
2
3
x
xx
xx
xx
3
)1)(13(
)1)(13(
)91(3 2
x
xx
xx
xxx
3
)1)(13(
)1)(13(
)31)(31(3
)13()31( xx
29 6 1x x
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Dividing Rational Expressions Rewrite as a multiplication problem
(Multiply the first by the reciprocal of the second.)
Factor numerators and denominators Simplify Multiply Rewrite the answers in simplified form
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Example:6
3
84
32
2
xx
xx
x
6
3
84
32
2
xx
xx
x )2)(3(
)3(
)2(4
3
xx
xx
x
)3(
)2)(3(
)2(4
3
xx
xx
x x4
3
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Multiplying or Dividing Rational Expressions by Polynomials Treat the denominator of the
polynomial expression as 1. Factor each numerator and
denominator Multiply/”Divide” Rewrite in simplified form
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Example xxx
xx
22
2
44
3108
1
4
4
3108 2
2
2 xx
x
xx
)4(
1
4
310822
2
xxx
xx
)14(
1
4
)32)(14(2
xxx
xxxx
x
24
1)32(34
32
x
x
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Assignment
Page 558 #16-49