Rational Rational ExpressionsExpressions

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Simplifying Rational Expressions

The objective is to be able to simplify a rational expression.

5

2x

3

92

x

x

Polynomial – The sum or difference of monomials.

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

Domain of a rational expression – the set of all real numbers except those for which the denominator is zero.

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.

5

5 7

x

x( )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 = 1

The x remains in the numerator.

The (x-7) remains in the denominator

5

5 7

x

x( ) x

x 7

x

x 7

Factoring the Numerator and Denominator

Factor the numerator.

Factor the denominator.

Divide out the common factors.

Write in simplified form.

3 9

1 2

2

3

x x

x

Factoring

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

3 9

1 2

2

3

x 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 9

1 2

2

3

x x

x

3 3

12 3

x x

x

( )

Factoring

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

3 9

1 2

2

3

x 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

3

x x

x

3 3

3 4 2

x x

x 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 2

x x

x x

( )

Step 5: Write in simplified form.

x

x

3

4 2

You Try It

Simplify the following rational expressions.

19

2 4

2

2.

x yz

xyz

23

4 32.

a

a a

33 1 5

7 1 02.

x

x x

42 1 5

1 2

2

2. x x

x x

51 4 3 5 2 1

1 2 3 0 1 8

2

2.

x x

x x

Restrictions on Rational Expressions

For what value of x is undefined? x x

x

2 2 1 5

4 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 5

x 5

The restriction is that x cannot equal 5.

YOU TRY ITWhat are the excluded values of the variables for the following rational expressions?

14 2

1 4

3 2

2 3.

y z

y z

23 6

2 1 2

2

. x

x

34 1 2

2 8

2

2.

c c

c c

Assignment

Problems 24 - 31