Algebra Examples

$f (x) = \frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$

Step 1

Factor $x^{3} - 2 x^{2} - 8 x$ .

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Step 1.1

Factor $x$ out of $x^{3} - 2 x^{2} - 8 x$ .

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Step 1.1.1

Factor $x$ out of $x^{3}$ .

$f (x) = \frac{x \cdot x^{2} - 2 x^{2} - 8 x}{3 x^{2} - 27}$

Step 1.1.2

Factor $x$ out of $- 2 x^{2}$ .

$f (x) = \frac{x \cdot x^{2} + x (- 2 x) - 8 x}{3 x^{2} - 27}$

Step 1.1.3

Factor $x$ out of $- 8 x$ .

$f (x) = \frac{x \cdot x^{2} + x (- 2 x) + x \cdot - 8}{3 x^{2} - 27}$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 2 x)$ .

$f (x) = \frac{x (x^{2} - 2 x) + x \cdot - 8}{3 x^{2} - 27}$

Step 1.1.5

Factor $x$ out of $x (x^{2} - 2 x) + x \cdot - 8$ .

$f (x) = \frac{x (x^{2} - 2 x - 8)}{3 x^{2} - 27}$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 1.2.1.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 1.2.1.2

Write the factored form using these integers.

$f (x) = \frac{x ((x - 4) (x + 2))}{3 x^{2} - 27}$

Step 1.2.2

Remove unnecessary parentheses.

$f (x) = \frac{x (x - 4) (x + 2)}{3 x^{2} - 27}$

Step 2

Factor $3 x^{2} - 27$ .

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Step 2.1

Factor $3$ out of $3 x^{2} - 27$ .

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Step 2.1.1

Factor $3$ out of $3 x^{2}$ .

$f (x) = \frac{x (x - 4) (x + 2)}{3 (x^{2}) - 27}$

Step 2.1.2

Factor $3$ out of $- 27$ .

$f (x) = \frac{x (x - 4) (x + 2)}{3 x^{2} + 3 \cdot - 9}$

Step 2.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 9$ .

$f (x) = \frac{x (x - 4) (x + 2)}{3 (x^{2} - 9)}$

Step 2.2

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

$f (x) = \frac{x (x - 4) (x + 2)}{3 (x^{2} - 3^{2})}$

Step 2.3

Factor.

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Step 2.3.1

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 $b = 3$ .

$f (x) = \frac{x (x - 4) (x + 2)}{3 ((x + 3) (x - 3))}$

Step 2.3.2

Remove unnecessary parentheses.

$f (x) = \frac{x (x - 4) (x + 2)}{3 (x + 3) (x - 3)}$

Step 3

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

No holes

Step 4