Review of Simplifying & Operations with Rational Expressions
Example 1
Simplify a Rational Expression
2
a. Simplify
2x + x – 6
5 x + 10
.
Since you cannot identify any common factors in the numerator and denominator, factor each of the
expressions.
2x
2
5x
x – 6
10
=
(2x – 3)(x
5(x
2)
Factor the numerator and denominator.
2)
1
=
(2 x – 3) ( x 2)
Divide the numerator and denominator by the common
5 ( x 2)
1
factor of (x + 2).
=
2x – 3
Simplify.
5
b. Under what conditions is this expression undefined?
Just as with a fraction, a rational expression is undefined if the denominator is equal to 0. To find when
this expression is undefined, completely factor the original denominator.
2x
2
5x
x – 6
10
=
(2x – 3)(x
5(x
2)
5x + 10 = 5( x + 2)
2)
The value that would make the denominator equal to 0 is –2. So, the expression is undefined when
x = –2.
Example 2
Use the Process of Elimination
Multiple–Choice Test Item
2
For what value(s) of a is
A. –5, 3
a + 2a – 15
2
a - 3a
B. –3, 5
undefined?
C. –5, 0, 3
D. 0, 3
Read the Test Item
You want to determine which values of a make the denominator equal to 0.
Solve the Test Item
Notice that a is in both terms of the expression in the denominator. That means that a = 0 will make the
denominator equal to 0. Therefore, you can eliminate choices A and B since they do not contain an
answer of 0. Since the denominator is a quadratic expression, it can have only 2 possible values. That
eliminates choice C.
Since 32 – 3(3) = 0, the answer is D.
Example 3
Simplify
Simplify by Factoring Out –1
2
4–b
2
b + 3b – 10
2
4 – b
b
2
3b – 10
.
2 – b 2
=
b
Factor the numerator and the denominator.
5)(b – 2)
(b
1
(–1) (b 2) (b 2)
=
2 – b = –(–2 + b) or –1(b – 2)
(b 5) (b 2)
1
(–1)(b
=
Example 4
(b
•
2
15 x
5x z
2xy
2
5x z
•
15x
=
4z
=
=
3 2
2
5b
3c
Simplify.
5
.
Factor.
5 • x • x • z• 2 •2• z
1
1 1 1
3 • y
Simplify.
2 • z • z
3y
6a c
5b
2
2
25a
b
1 1
1 1
2 • x • y •3• 5 • x
Simplify.
2
3 2
6a c
–b – 2
4z
2z
b. Simplify
5)
or
Multiply and Divide Rational Expressions
2 xy
a. Simplify
2)
3c
÷
2
25a
.
3 2
=
6a c
5b
2
•
25a
3c
1
=
=
=
Multiply by the reciprocal of the divisor.
2
1
1
1
2• 3 •a •a •a • c • c • 5 •5•a
5 •b•b• 3 • c • c
1
1 1 1
2 • 5• a • a• a • a
b • b
10a 4
b
2
Factor.
Simplify.
Simplify.
Example 5
Polynomials in the Numerator and Denominator
Simplify each expression.
a.
x 2 x 2 4x 8
•
x 2 5x 6 x 1
x 2 x 2 4x 8
•
x 2 5x 6 x 1
1
1
(x 2)(x 1) 4(x 2)
=
•
(x 2)(x 3) x 1
1
4(x
x
4x
=
x
=
b.
3x
3x + 6
2 x – 10
6
2x – 10
÷
Factor.
1
2)
3
8
3
Simplify.
2
÷
4x + 7x – 2
2
2 x – 7 x – 15
4x
2x
2
2
7x – 2
– 7x – 15
.
=
3x
6
2x – 10
•
1
=
3 ( x 2)
•
1
=
=
3)
2(4x – 1)
6x 9
8x 2
2
– 7x – 15
2
7x – 2
4x
Multiply by the reciprocal of the divisor.
1
2 ( x 5)
3(2x
2x
(2 x 3) ( x 5)
( x 2) (4 x – 1)
1
Factor.
Simplify.
Example 6
Simplify a Complex Fraction
z –1
Simplify
2
z –1
.
2
z – 2z + 1
2
z –z–2
z – 1
2
z – 2z
1
z – 1
z2 – 1
= 2
÷ 2
2
z – 2z
1
z – z – 2
z – 1
z
2
Express as a division expression.
– z – 2
=
z – 1
•
2
z – 1
z
z
2
2
1
=
( z 1)
( z 1) ( z 1)
1
1
=
z – 2
(z – 1)
2
– z – 2
– 2z
Multiply by the reciprocal of the divisor.
1
1
•
( z – 2) ( z 1)
( z – 1)( z – 1)
Factor.
Simplify.