Algebra Examples

$f (x) = \frac{x^{2} - 4}{x^{2} - 9}$

Step 1

Factor $x^{2} - 4$ .

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Step 1.1

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 2

Factor $x^{2} - 9$ .

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$f (x) = \frac{(x + 2) (x - 2)}{x^{2} - 3^{2}}$

Step 2.2

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 $b = 3$ .

$f (x) = \frac{(x + 2) (x - 2)}{(x + 3) (x - 3)}$

Step 3

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

No holes

Step 4