Reducing Rational Expressions to Lowest Terms
In the next example we use the techniques for factoring polynomials.
Example 1
Reducing
Reduce each rational expression to its lowest terms.
Solution
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Factor. |
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Factor completely. |
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Divide out the common factors. |
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Factoring out -4 will give the
common factor a - 2. |
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Difference of two cubes,
difference of two squares |
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Divide out common factors. |
The rational expressions in Example 1(a) are equivalent because they have the
same value for any replacement of the variables, provided that the replacement is in
the domain of both expressions. In other words, the equation
is an identity. It is true for any value of x except 2 and -3.
The main points to remember for reducing rational expressions are summarized
as follows.
Strategy for Reducing Rational Expressions
1. All reducing is done by dividing out common factors.
2. Factor the numerator and denominator completely to see the common
factors.
3. Use the quotient rule to reduce a ratio of two monomials involving
exponents.
4. We may have to factor out a common factor with a negative sign to get
identical factors in the numerator and denominator.
Helpful hint
Since -1(a - b)
= b - a, placement of a negative
sign in a rational expression
changes the appearance of
the expression:
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