11-2 Rational Expressions

Algebra 1 Glencoe McGraw-Hill Linda Stamper

A rational number is a number that can be written as the

quotient of two integers, such as.

17 ,

34

,21

4x3 9x

2x2 1x

13x2

3x

2x2x2

Because a rational expression involves division, the denominator may not equal zero. Any values of a variable that result in a denominator of zero must be excluded from the domain of that variable. These are called excluded values of the rational expression.

A fraction whose numerator and denominator are nonzero polynomials is a rational expression.

Excluded values must always be determined from the original denominator.

State the excluded value for the rational expression.

6x5x

5x2

2

6m35m

To find the excluded value set the

denominator equal to zero and solve.06m

6m Excluded value is 6.

06x5x2 6

-5

-2 –3 03x2x

Set the denominator equal to zero. Factor and solve.

03xor02x 3xor2x Excluded values are 2

and 3.

State the excluded value for each rational expression.

63x13x

16x2

2

x35x61

0x3 0x

0

063x13x2 36

-13

-9 –4 04x9x 04xor09x

4xor9x

9, 4

Example 1

Example 2

Example 3

Example 4

9x

1x2

2

09x2 03x3x

-3, 3

2x53x

4x32

02x5x3 2 6

5 3 2

1x3 02x33x3

02x31x 02x3or01x

32

,1

Can you factor out a greatest monomial

factor?

Simplifying a rational expression is similar to simplifying fractions because the variables in a rational expression represent real numbers.

To understand today’s lesson, you MUST know the difference between factors and terms.

Factors are values that are multiplied.

32

Terms are values that are added or subtracted.32 32

32

To simplify a rational expression, factor the numerator and denominator and then divide out any common factors. A rational number is in simplest form if its numerator and denominator have no factors in common other than 1.

Simplify the rational expression if possible.

2424

1 xx 1

x2x2

1

x4x4

1x

x4 simplified

xx4

You can cancel (divide out)

factors but not terms.

Simplify the expression. State the excluded value of the variable.

The problem.x2x36 2

Reduce the coefficients.Reduce the variables.

x18

Simplify.

18 1

State the excluded value.

0

Answer line

;x18

0

18a36a

6a36a

31

1

Simplify the expression. State the excluded value of the variable.

Factor out a GCF.

State the excluded value.

6

Excluded values must always be determined from the original denominator.

Simplify the rational expression. State the excluded value of the variable.

3x

1x3x

Factor.3x

32xx2

1x

1

3

23 1

State the excluded value.

3

Simplify the rational expression. State the excluded values of the variable.

340x

5x28x

1

1

82

Ex. 5 Ex. 6 x5

x35 x

x3 //

1

xx3x

3x

Ex. 7 5xx2x2

5xx2

/

/

1

Remember you can cancel (divide out)

factors but not terms.

Ex. 83x3x5

simplified

3x3x5

Excluded value is 0. Excluded value is

0.

Excluded values are 0 and -5.

Excluded value is -3.

Answer line

;5x

x2

0, -5

Simplify the rational expression. State the excluded values of the variable.

x105x15

Example 10

5x103xx2

Example 9

12b8b

36b2

2

Example 11

Example 12

28x3x

21x10x2

2

15x13x2

20x9x2

2

Example 13

Example 14

x8x2

x48x32

3

Simplify the rational expression. State the excluded values of the variable.

x105x15

Factor out a GCF.

x215x15

x21x3

//

3

1x2x3

Example 10

5x103xx2

5x2x5x

2x

1

10

35 2

Example 9

Excluded value is ½.

Excluded value is 5.

Excluded values must always be determined from the original denominator.

12b8b

36b2

2

2b6b6b6b

2b6b

1

Simplify the rational expression. State the excluded values of the variable.

Example 11

Example 12

Excluded value are 6 and 2.

28x3x

21x10x2

2

4x7x3x7x

4x3x

1

Excluded value are 7 and -4.

Notice there are no

parentheses in this answer!

15x13x2

20x9x2

2

5x3x24x5x

3x24x

1

Notice there are no

parentheses in this answer!

Simplify the rational expression. State the excluded values of the variable.

Example 13

Excluded value are .

5,23

Example 14

x8x2

x48x32

3

4xx216xx3 2

24x3

1

Leave the answer in

factor form!

4xx2

4x4xx3

1

Excluded value are 0 and 4.

11-A2 Pages 587-588 # 15–17,24–39,57-60.