Algebra - Rational Expressions

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Section 1.6 : Rational Expressions

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11. Perform the indicated operation in the following expression.

\[\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}}\]

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Start Solution

We first need the least common denominator for this rational expression. However, before we get that we’ll need to factor the denominator of the first term. Doing this gives,

\[\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}} = \frac{1}{{\left( {x - 6} \right)\left( {x - 7} \right)}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}}\] Show Step 2

The least common denominator is then,

\[{\mbox{lcd : }}\left( {x - 6} \right)\left( {x - 7} \right)\]

Remember that we only take the highest power on each term in the denominator when setting up the least common denominator.

Show Step 3

Next, multiply each term by an appropriate quantity to get the least common denominator into the denominator of each term.

\[\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}} = \frac{1}{{\left( {x - 6} \right)\left( {x - 7} \right)}} + \frac{{\left( {x + 1} \right)\left( {x - 7} \right)}}{{\left( {x - 6} \right)\left( {x - 7} \right)}} - \frac{{{x^2}\left( {x - 6} \right)}}{{\left( {x - 7} \right)\left( {x - 6} \right)}}\] Show Step 4

Finally, all we need to do is the addition and subtraction then simplify the numerator of the result.

\[\begin{align*}\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}} & = \frac{{1 + \left( {x + 1} \right)\left( {x - 7} \right) - {x^2}\left( {x - 6} \right)}}{{\left( {x - 6} \right)\left( {x - 7} \right)}}\\ & = \frac{{1 + {x^2} - 6x - 7 - {x^3} + 6{x^2}}}{{\left( {x - 6} \right)\left( {x - 7} \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{ - {x^3} + 7{x^2} - 6x - 6}}{{\left( {x - 6} \right)\left( {x - 7} \right)}}}}\end{align*}\]