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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.