Algebra Examples

$f (x) = \frac{x^{2} - 81}{x^{2} + 15 x + 56}$

Step 1

Factor $x^{2} - 81$ .

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Step 1.1

Rewrite $81$ as $9^{2}$ .

$f (x) = \frac{x^{2} - 9^{2}}{x^{2} + 15 x + 56}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$f (x) = \frac{(x + 9) (x - 9)}{x^{2} + 15 x + 56}$

Step 2

Factor $x^{2} + 15 x + 56$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $56$ and whose sum is $15$ .

$7, 8$

Step 2.2

Write the factored form using these integers.

$f (x) = \frac{(x + 9) (x - 9)}{(x + 7) (x + 8)}$

Step 3

Since no factors can be removed from the denominator, there are no holes in the graph.

No holes

Step 4