Rational Expressions

Simplifying

Polynomial – The sum or difference of monomials.

Rational expression – A fraction whose numerator and denominator are polynomials.

Reduced form – a rational expression in which the numerator and denominator have no factors in common.

Divide out the common factors

Factor the numerator and denominator and then divide the common factors

Dividing Out Common Factors

Step 1 – Identify any factors which are common to both the numerator and the denominator.

55 7

xx( )

The numerator and denominator have a common factor.The common factor is the five.

Dividing Out Common FactorsStep 2 – Divide out the common factors.The fives can be divided since 5/5 = 1The x remains in the numerator.The (x-7) remains in the denominator

55 7

xx( )

xx 7

Factoring the Numerator and DenominatorFactor the numerator.Factor the denominator.Divide out the common factors.Write in simplified form.

3 91 2

2

3x x

x

FactoringStep 1: Look for common factors to both terms in the numerator.

3 91 2

2

3x x

x 3 is a factor of both 3 and

9.X is a factor of both x2 and x.Step 2: Factor the numerator.

3 91 2

2

3x x

x 3 3

12 3x xx

( )

FactoringStep 3: Look for common factors to the terms in the denominator and factor.

3 91 2

2

3x x

x

The denominator only has one term. The 12 and x3 can be factored. The 12 can be factored into 3 x 4.The x3 can be written as x • x2.3 9

1 2

2

3x x

x 3 3

3 4 2x xx x

( )

Divide and Simplify

Step 4: Divide out the common factors. In this case, the common factors divide to become 1.3 3

3 4 2x x

x x( )

Step 5: Write in simplified form.

xx

34 2

Example

92 4

2

2x yzxyz

Problem 19

24

2

2x yzxyz

3 33 8

x x y zx y z z

33 x y zx y z

38xz

1 38xz

38xz

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

Example 2

aa a

34 32

Problem 2a

a a

3

4 32 aa a

33 1( )( )

Factor the numerator and denominator

aa

33

11a

1 11a 1

1a

Divide out the common factors.

Write in simplified form.

You Try It

Example 3

3 1 57 1 02x

x x

Problem 33 15

7 102x

x x

3 55 2

( )( )( )

xx x

xx

55

32x

1 3

2x 3

2x

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

You Try It

Example 4

x xx x

2

22 1 5

1 2

Problem 4

You Try It

x xx x

2

22 15

12 ( )( )

( )( )x xx x

5 34 3

xx

33

xx

54

1 xx

54 x

x

54

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

Restrictions on Rational Expressions

For what value of x is undefined? x xx

2 2 1 54 2 0

It is undefined for any value of “x” which makes the denominator zero.

4 2 0 0x 4 5 0( )x x 5 0x 5 5 0 5x 5

The restriction is that x cannot equal 5.

Example 5

14 3 5 211 2 3 0 1 8

2

2x xx x

Problem 514 3 5 2 112 3 0 18

2

2x xx x

7 2 5 36 2 6 3

2

2( )( )x xx x

( )( )

2 5 32 6 3

2

2x xx x

76

1 76

76

You Try It

Factor the numerator and denominator

Divide out the common factors.

Write in simplified form.

123 :Simplify 2

2

yyy

11

21yyyy

12

yy

yy48

2412 :Simplify

yy412

212

yy4

2

xxxx

232 :Simplify 2

2

2 13 2

x xx x

2 13 2xx

2

2

9Simplify: 2 15a

a a

3 35 3

a aa a

121

aa

xx

44 :Simplify

41

4

x

x

11

1

Simplifying4 3

5 5x xx 2

525

qq

3x

3

5x

5q

15q

1x 5 1x 5q 5q

12.1 – Simplifying Rational Expressions

Simplifying

2

2

11 182

x xx x

2x

91

xx

9x

2x 1x

12.1 – Simplifying Rational Expressions

Simplifying

44xx

4x

4x

4x

1 4x 1

12.1 – Simplifying Rational Expressions