3.8 Simplify Rational ExpressionsVocabulary An expression that can be written as a ratio of two...

3.8 Simplify Rational ExpressionsVocabularyVocabulary

An expression that can be written as a ratio of two polynomials.

Rational Expression

A number that makes a rational expression undefined.Excluded Value

A rational expression when the numerator and denominator have no factors in common other than 1.

Simplest form of a rational expression

3.8 Simplify Rational Expressions

Example 1 Find excluded valuesFind excluded values

84 .

xx

a

Find the excluded values, if any, of the expression.Find the excluded values, if any, of the expression.

16

3 .

2 x

xb

2

1 .

2 xc

a. The expression is undefined when ________ = 0,84 x

x 84 x

or x = ___, the excluded value is _____.2 2

3.8 Simplify Rational Expressions

Example 1 Find excluded valuesFind excluded values

84 .

xx

a

Find the excluded values, if any, of the expression.Find the excluded values, if any, of the expression.

16

3 .

2 x

xb

2

1 .

2 xc

b. The expression is undefined when ________ = 0,16

32 x

x162 x

or ______)(_______) = 0.4x 4xThe solutions to the of the equation are ____ and ____.4 4The excluded values are ____ and ____.4 4

3.8 Simplify Rational Expressions

Example 1 Find excluded valuesFind excluded values

84 .

xx

a

Find the excluded values, if any, of the expression.Find the excluded values, if any, of the expression.

16

3 .

2 x

xb

2

1 .

2 xc

c. The expression is undefined when ________ = 0.2

12 x

22 x

The graph of y = x2 + 2 ________________________.

So, the quadratic equation has _____________.

does not cross the x-axis

no real roots

There are ___________________.no excluded values

3.8 Simplify Rational ExpressionsCheckpoint. Find the excluded values, if any, of the Checkpoint. Find the excluded values, if any, of the expression. expression.

x

x

14

6 .1

014 x14 14x 0

20

19 .2

2 xx

x

0202 xx

0 x x5 405 x 04 x

5x

55

4x

44

3.8 Simplify Rational Expressions

Example 2 Simplify expressions by dividing out monomialsSimplify expressions by dividing out monomials

26

18 .x

xa

Simplify the rational expression, if possible. State the Simplify the rational expression, if possible. State the excluded values.excluded values.

The excluded value is _____.0

6 3 x

6 x x

3x

06 2 x6 6

x 002 x

3.8 Simplify Rational Expressions

Example 2 Simplify expressions by dividing out monomialsSimplify expressions by dividing out monomials

x

xxb

24

612 .

2

Simplify the rational expression, if possible. State the Simplify the rational expression, if possible. State the excluded values.excluded values.

The excluded value is _____.0

6 x

6 4 x

12 x

4

024 x24 24x 0

x2 1

6 x

6 4 xx2 1

3.8 Simplify Rational ExpressionsCheckpoint. Simplify the rational expression, if Checkpoint. Simplify the rational expression, if possible. State the excluded values. possible. State the excluded values.

35

7 .3

x

035 x

5

x5

3

Since the expression on the top and bottom cannot be factored, and they do not have any common factors other than 1, the expression is already simplified.

3 3x5 3

5

3.8 Simplify Rational ExpressionsCheckpoint. Simplify the rational expression, if Checkpoint. Simplify the rational expression, if possible. State the excluded values. possible. State the excluded values.

255

5 .4

2 x

x

0255 2 x

5

25 2525x 25

5

5 x

5 2x 5

x52 x

52 x

x 5

3.8 Simplify Rational ExpressionsCheckpoint. Simplify the rational expression, if Checkpoint. Simplify the rational expression, if possible. State the excluded values. possible. State the excluded values.

42

6 .5

3

xx

042 x

2

4 4x2 4

2

2 3x

2 x 2

33x2x

2x

3

3.8 Simplify Rational Expressions

Example 3 Simplify expressions by dividing out binomialsSimplify expressions by dividing out binomials

65

122

2

xx

xx

Simplify . State the excluded values.Simplify . State the excluded values.

The excluded values are _____ and _____.3

4x 3x3x 2x

4x

2x

023 xx

65

122

2

xx

xx

03 x 02 x3x

33

2x

22

2

3.8 Simplify Rational ExpressionsCheckpoint. Simplify the rational expression, if Checkpoint. Simplify the rational expression, if possible. State the excluded values. possible. State the excluded values.

183

67 .6

2

2

xx

xx

036 xx

6x 1x

6x 3x

1x3x

06 x 03 x6x

66

3x

33

3.8 Simplify Rational ExpressionsCheckpoint. Simplify the rational expression, if Checkpoint. Simplify the rational expression, if possible. State the excluded values. possible. State the excluded values.

145

4 .7

2

2

xx

x

027 xx

2x 2x 7x 2x

2x

7x

07 x 02 x7x

77

2x

22

3.8 Simplify Rational Expressions

Pg. 178, 3.8 #1-21