Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational...
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![Page 1: Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are.](https://reader034.fdocuments.net/reader034/viewer/2022051717/5a4d1b5f7f8b9ab0599acc68/html5/thumbnails/1.jpg)
Rational Expressions
Simplifying
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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.
![Page 3: Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are.](https://reader034.fdocuments.net/reader034/viewer/2022051717/5a4d1b5f7f8b9ab0599acc68/html5/thumbnails/3.jpg)
Divide out the common factors
Factor the numerator and denominator and then divide the common factors
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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.
![Page 5: Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are.](https://reader034.fdocuments.net/reader034/viewer/2022051717/5a4d1b5f7f8b9ab0599acc68/html5/thumbnails/5.jpg)
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
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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
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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
( )
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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
( )
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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
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Example
92 4
2
2x yzxyz
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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.
![Page 12: Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are.](https://reader034.fdocuments.net/reader034/viewer/2022051717/5a4d1b5f7f8b9ab0599acc68/html5/thumbnails/12.jpg)
Example 2
aa a
34 32
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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
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Example 3
3 1 57 1 02x
x x
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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
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Example 4
x xx x
2
22 1 5
1 2
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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.
![Page 18: Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are.](https://reader034.fdocuments.net/reader034/viewer/2022051717/5a4d1b5f7f8b9ab0599acc68/html5/thumbnails/18.jpg)
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.
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Example 5
14 3 5 211 2 3 0 1 8
2
2x xx x
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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.
![Page 21: Rational Expressions Simplifying. Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are.](https://reader034.fdocuments.net/reader034/viewer/2022051717/5a4d1b5f7f8b9ab0599acc68/html5/thumbnails/21.jpg)
123 :Simplify 2
2
yyy
11
21yyyy
12
yy
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yy48
2412 :Simplify
yy412
212
yy4
2
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xxxx
232 :Simplify 2
2
2 13 2
x xx x
2 13 2xx
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2
2
9Simplify: 2 15a
a a
3 35 3
a aa a
121
aa
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xx
44 :Simplify
41
4
x
x
11
1
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Simplifying4 3
5 5x xx 2
525
3x
3
5x
5q
15q
1x 5 1x 5q 5q
12.1 – Simplifying Rational Expressions
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Simplifying
2
2
11 182
x xx x
2x
91
xx
9x
2x 1x
12.1 – Simplifying Rational Expressions
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Simplifying
44xx
4x
4x
4x
1 4x 1
12.1 – Simplifying Rational Expressions