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Section 1.6 : Rational Expressions
7. Perform the indicated operation in the following expression and reduce the answer to lowest terms.
\[\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}}\]Show All Steps Hide All Steps
Start SolutionThis is just a division and so let’s first convert it to a product.
\[\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}} = \frac{3}{{x + 1}}\,\centerdot \,\frac{{{x^2} + 11x + 10}}{{x + 4}}\] Show Step 2Now we can factor each of the second term as much as possible to get,
\[\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}} = \frac{3}{{x + 1}}\,\centerdot \,\frac{{\left( {x + 1} \right)\left( {x + 10} \right)}}{{x + 4}}\] Show Step 3Now cancel as much as possible to reduce to lowest terms and do the product.
\[\frac{{\displaystyle \frac{3}{{x + 1}}}}{{\displaystyle \frac{{x + 4}}{{{x^2} + 11x + 10}}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{3\left( {x + 10} \right)}}{{x + 4}}}}\]