Paul's Online Notes
Paul's Online Notes
Home / Algebra / Preliminaries / Rational Expressions
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 1-6 : Rational Expressions

11. Perform the indicated operation in the following expression.

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

Show All Steps Hide All Steps

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.

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