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### Section 1-6 : Rational Expressions

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

$\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}}$

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So, we first need to do is convert this into a product.

$\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}} = \frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}}\,\centerdot \,\frac{{{x^2} + 7x + 6}}{{{x^2} - x - 42}}$

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,

$\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}} = \frac{{\left( {x - 7} \right)\left( {x + 7} \right)}}{{\left( {2x - 5} \right)\left( {x + 1} \right)}}\centerdot \frac{{\left( {x + 1} \right)\left( {x + 6} \right)}}{{\left( {x - 7} \right)\left( {x + 6} \right)}}$ Show Step 3

Finally cancel as much as possible to reduce to lowest terms and do the product.

$\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{x + 7}}{{2x - 5}}}}$