To simplify rational expressions we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.
Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.
Note that removing the
x ’s from
would be like cancelling the 2’s in the fraction
!
We now summarize the steps you should follow to simplify rational expressions.
Simplify a rational expression.
- Factor the numerator and denominator completely.
- Simplify by dividing out common factors.
Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors!
We’ll use the methods we covered in
Factoring to factor the polynomials in the numerators and denominators in the following examples.
Simplify:
Solution
Can you tell which values of
x must be excluded in this example?
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Simplify:
Solution
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Simplify:
Solution
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Simplify:
Solution
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