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### Section 1-6 : Rational Expressions

6. Perform the indicated operation in the following expression and reduce the answer to lowest terms.

$\frac{{{x^2} - 2x - 8}}{{2{x^2} - 8x - 24}} \div \frac{{{x^2} - 9x + 20}}{{{x^2} - 11x + 30}}$

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Start Solution

So, 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 2

Now 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 3

Finally 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.