Canceling Like Factors
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.
Polynomial / Monomial
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
| 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 bereduced if possible:
· 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. |
- Polynomials must be factored first. You can’t cancel factors unless you can see the factors:
Example:
- Notice how canceling the (x– 2) from the denominator left behind a factor of 1
Multiplication and Division
Same rules as for rational numbers!
Multiplication
- Both the numerators and the denominators multiply together
- Common factors may be cancelled before multiplying
Example:
Given Equation: |
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| First factor all the expressions: (I also put the denominators in parentheses because then it is easier t 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 outthe factors, just leave them as shown.  |
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
Addition and Subtraction
Same procedure as for rational numbers!
- Only the numerators can be addedtogether, and only when 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 commondenominator, but not necessarily the LCD (the final answer may have to be reduced).
Example:
Given equation: |
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Factor both denominators: |
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Assemble the LCD: Note that the LCD contains bothdenominators   |
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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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