1Cademy - Example of Simplifying a Rational Expression with Opposite Factors

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Opposites in a Rational Expression

Example

Example of Simplifying a Rational Expression with Opposite Factors

To simplify a rational expression that contains factors that are opposites, first factor the numerator and denominator completely. Then, identify any factors that are opposites and cancel them, which introduces a factor of $-1$ .

For example, to simplify $\frac{x^2 - 4x - 32}{64 - x^2}$ :

Factor the numerator and the denominator to get $\frac{(x - 8)(x + 4)}{(8 - x)(8 + x)}$ .
Recognize that the binomial factors $x - 8$ and $8 - x$ are opposites.
Cancel the opposite factors, replacing them with $-1$ .
Apply the $-1$ to the remaining expression to obtain the final simplified result: $-\frac{x + 4}{x + 8}$ .