Section 1.6 : Rational Expressions
6. Perform the indicated operation in the following expression and reduce the answer to lowest terms.
\[\frac{{{x^2} - 2x - 8}}{{2{x^2} - 8x - 24}} \div \frac{{{x^2} - 9x + 20}}{{{x^2} - 11x + 30}}\]Show All Steps Hide All Steps
Start SolutionSo, we first need to do is convert this into a product.
\[\frac{{{x^2} - 2x - 8}}{{2{x^2} - 8x - 24}} \div \frac{{{x^2} - 9x + 20}}{{{x^2} - 11x + 30}} = \frac{{{x^2} - 2x - 8}}{{2{x^2} - 8x - 24}}\,\centerdot \,\frac{{{x^2} - 11x + 30}}{{{x^2} - 9x + 20}}\]Make sure that you don’t do the factoring and canceling until you’ve converted the division to a product.
Show Step 2Now we can factor each of the terms as much as possible to get,
\[\frac{{{x^2} - 2x - 8}}{{2{x^2} - 8x - 24}} \div \frac{{{x^2} - 9x + 20}}{{{x^2} - 11x + 30}} = \frac{{\left( {x - 4} \right)\left( {x + 2} \right)}}{{2\left( {x + 2} \right)\left( {x - 6} \right)}}\,\centerdot \,\frac{{\left( {x - 5} \right)\left( {x - 6} \right)}}{{\left( {x - 5} \right)\left( {x - 4} \right)}}\] Show Step 3Finally cancel as much as possible to reduce to lowest terms and do the product.
\[\frac{{{x^2} - 2x - 8}}{{2{x^2} - 8x - 24}} \div \frac{{{x^2} - 9x + 20}}{{{x^2} - 11x + 30}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{2}}}\]Don’t worry if all the variables end up cancelling out after you are done reducing to lowest terms. It will happen on occasion so don’t worry about it when it does.