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### Section 1-6 : Rational Expressions

10. Perform the indicated operation in the following expression.

$\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}}$

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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,

$\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}} = \frac{x}{{{{\left( {x + 6} \right)}^2}}} - \frac{{x - 8}}{{x + 6}}$ Show Step 2

The least common denominator is then,

${\mbox{lcd : }}{\left( {x + 6} \right)^2}$

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.

$\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}} = \frac{x}{{{{\left( {x + 6} \right)}^2}}} - \frac{{\left( {x - 8} \right)\left( {x + 6} \right)}}{{\left( {x + 6} \right)\left( {x + 6} \right)}}$ Show Step 4

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

$\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}} = \frac{{x - \left( {x - 8} \right)\left( {x + 6} \right)}}{{{{\left( {x + 6} \right)}^2}}} = \frac{{x - \left( {{x^2} - 2x - 48} \right)}}{{{{\left( {x + 6} \right)}^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{48 + 3x - {x^2}}}{{{{\left( {x + 6} \right)}^2}}}}}$