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Divide out the common factors Factor the numerator and denominator and then divide the common factors
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Rational Expressions
Simplifying
Polynomial – The sum or difference of monomials.
Rational expression – A fraction whose numerator and denominator are polynomials.
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.
55 7
xx( )
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 = 1The x remains in the numerator.The (x-7) remains in the denominator
55 7
xx( )
xx 7
Factoring the Numerator and DenominatorFactor the numerator.Factor the denominator.Divide out the common factors.Write in simplified form.
3 91 2
2
3x x
x
FactoringStep 1: Look for common factors to both terms in the numerator.
3 91 2
2
3x 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 91 2
2
3x x
x 3 3
12 3x xx
( )
FactoringStep 3: Look for common factors to the terms in the denominator and factor.
3 91 2
2
3x 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
3x x
x 3 3
3 4 2x xx 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 2x x
x x( )
Step 5: Write in simplified form.
xx
34 2
Example
92 4
2
2x yzxyz
Problem 19
24
2
2x yzxyz
3 33 8
x x y zx y z z
33 x y zx y z
38xz
1 38xz
38xz
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
Example 2
aa a
34 32
Problem 2a
a a
3
4 32 aa a
33 1( )( )
Factor the numerator and denominator
aa
33
11a
1 11a 1
1a
Divide out the common factors.
Write in simplified form.
You Try It
Example 3
3 1 57 1 02x
x x
Problem 33 15
7 102x
x x
3 55 2
( )( )( )
xx x
xx
55
32x
1 3
2x 3
2x
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
You Try It
Example 4
x xx x
2
22 1 5
1 2
Problem 4
You Try It
x xx x
2
22 15
12 ( )( )
( )( )x xx x
5 34 3
xx
33
xx
54
1 xx
54 x
x
54
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
Restrictions on Rational Expressions
For what value of x is undefined? x xx
2 2 1 54 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 5x 5
The restriction is that x cannot equal 5.
Example 5
14 3 5 211 2 3 0 1 8
2
2x xx x
Problem 514 3 5 2 112 3 0 18
2
2x xx x
7 2 5 36 2 6 3
2
2( )( )x xx x
( )( )
2 5 32 6 3
2
2x xx x
76
1 76
76
You Try It
Factor the numerator and denominator
Divide out the common factors.
Write in simplified form.
123 :Simplify 2
2
yyy
11
21yyyy
12
yy
yy48
2412 :Simplify
yy412
212
yy4
2
xxxx
232 :Simplify 2
2
2 13 2
x xx x
2 13 2xx
2
2
9Simplify: 2 15a
a a
3 35 3
a aa a
121
aa
xx
44 :Simplify
41
4
x
x
11
1
Simplifying4 3
5 5x xx 2
525
3x
3
5x
5q
15q
1x 5 1x 5q 5q
12.1 – Simplifying Rational Expressions
Simplifying
2
2
11 182
x xx x
2x
91
xx
9x
2x 1x
12.1 – Simplifying Rational Expressions
Simplifying
44xx
4x
4x
4x
1 4x 1
12.1 – Simplifying Rational Expressions