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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
