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MTH 092
Section 12.1Simplifying Rational Expressions
Section 12.2Multiplying and Dividing Rational Expressions
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Fractions Again?!?!?!?
• A rational expression is of the form
Where P and Q are polynomials with Q not equal to 0.
Q
P
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Why Can’t Q be equal to 0?
• Recall, from our work with slope, that division by 0 is undefined.
• If Q is a polynomial, then it has variables.• Those variables cannot take on values that
cause Q to become 0.• To figure out what those values are, set Q = 0
and solve.• Key words: undefined, domain
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Examples
• Find any numbers for which each rational expression is undefined:
145
111
219
25
15
3
2
2
2
3
xx
x
xx
x
x
xx
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Reducing To Lowest Terms
• Recall that reducing a fraction means dividing the numerator and denominator by the same value (usually the greatest common factor).
• Reducing a rational expression involves two steps:
1.Factor both the numerator and denominator.2.Cancel common factors.• Cancel factors, not terms.
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Examples: Reduce to Lowest Terms
592
144
7017
10
11
99925
5
9
9
2
2
2
34
2
2
xx
xx
xx
xx
xx
xx
x
y
y
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Multiplying and Dividing
1. Factor the numerators and denominators completely.
2. Cancel common factors.3. Remember that when you are dividing, you
must multiply by the reciprocal of the second rational expression.
4. You can not cancel terms. You can not cancel parts of terms.
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Multiply or Divide
32
10020
352
13512
2811
4415
209
25
155
5
3
6
5
20
244
2
2
2
2
2
2
x
x
xx
xxx
xx
xx
xx
xx
xx
x
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Opposite Factors Make -1
• For all real numbers a and b,
• The -1 is usually put in the numerator and can be distributed.
1ab
ba
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Apply “the rule of -1”
• Reduce, multiply, or divide as indicated:
9
12
3
327
339
7
49
9
9
2
222
5
2
x
xy
x
yx
ab
ba
x
y
y
y
y