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When we reduce a common fraction such as
we do so by noticing that there is a factor common to both the numerator and
the denominator (a factor of 2 in this example), which we can divide out of
both the numerator and the denominator.
We use exactly the same procedure to reduce rational expressions.
Each term in the numerator must have a factor that cancels a common
factor in the denominator.
 ,
but
cannot be reduced because the 2 is not a common factor of the entire
numerator.
WARNING You can only cancel a factor of
the entire numerator with a factor of the entire denominator
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However, as an alternative, a fraction with more
than one term in the numerator can be split up into separate fractions with
each term over the same denominator; then each separate fraction can be
reduced if possible:
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Think of this as the reverse of adding fractions over
a common denominator.
Sometimes this is a useful thing to do, depending on the circumstances.
You end up with simpler fractions, but the price you pay is that you have
more fractions than you started with.
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Polynomials must be factored
first. You can’t cancel factors unless you can see the factors:
Example:
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Notice how canceling the (x – 2) from
the denominator left behind a factor of 1
Same rules as for rational numbers!
Multiplication
- Both the numerators and
the denominators multiply together
- Common factors may be
cancelled before multiplying
Example:
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Given Equation:
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First factor all the expressions:
(I also put the denominators in parentheses because then it is easier to see
them as distinct factors)
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Now cancel common factors—any factor
on the top can cancel with any factor on the bottom:
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Now just multiply what’s left.
You usually do not have to multiply out the factors, just leave them as
shown.
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Division
- Multiply by the
reciprocal of the divisor
- Invert the second
fraction, then proceed with multiplication as above
- Do not attempt to cancel
factors before it is written as a multiplication
Same procedure as for rational numbers!
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Only the numerators can add
together, once all the denominators are the same
Finding the LCD
- The
LCD is built up of all the factors of
the individual denominators, each factor
included the most number of times it
appears in an individual denominator.
- The product of all the denominators is always a common denominator,
but not necessarily the LCD (the final
answer may have to be reduced).
Example:
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Given equation:
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Factor both denominators:
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Assemble the LCD:
Note that the LCD contains both denominators
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Build up the fractions so that they
both have the LCD for a denominator:
(keep both denominators in factored form to make it easier to see what
factors they need to look like the LCD)
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Now that they are over the same
denominator, you can add the numerators:
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And simplify:
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