Algebra - Rational Expressions

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Section 1.6 : Rational Expressions

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3. Reduce the following rational expression to lowest terms.

\[\frac{{2{x^2} - x - 28}}{{20 - x - {x^2}}}\]

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Start Solution

First, we need to factor the numerator and denominator as much as we can. Doing that gives,

\[\frac{{2{x^2} - x - 28}}{{20 - x - {x^2}}} = \frac{{2{x^2} - x - 28}}{{ - \left( {{x^2} + x - 20} \right)}} = \frac{{\left( {2x + 7} \right)\left( {x - 4} \right)}}{{ - \left( {x + 5} \right)\left( {x - 4} \right)}}\]

Notice that in order to make factoring the denominator somewhat easier we first factored a minus sign out of the denominator.

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Now all we need to do is cancel all the factors that we can in order to reduce the rational expression to lowest terms.

\[\frac{{2{x^2} - x - 28}}{{20 - x - {x^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{2x + 7}}{{x + 5}}}}\]

Recall that the minus sign in the denominator can be put out in front of the rational expression if we choose to put it there (as we did here).