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Section 1.6 : Rational Expressions
3. Reduce the following rational expression to lowest terms.
\[\frac{{2{x^2} - x - 28}}{{20 - x - {x^2}}}\]Show All Steps Hide All Steps
Start SolutionFirst, 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.
Show Step 2Now 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).