Precalculus Examples

$\frac{x^{3} + 5 x^{2} + 6 x}{x^{2} - 4}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x$ out of $x^{3} + 5 x^{2} + 6 x$ .

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Step 1.1.1

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

$\frac{x \cdot x^{2} + 5 x^{2} + 6 x}{x^{2} - 4}$

Step 1.1.2

Factor $x$ out of $5 x^{2}$ .

$\frac{x \cdot x^{2} + x (5 x) + 6 x}{x^{2} - 4}$

Step 1.1.3

Factor $x$ out of $6 x$ .

$\frac{x \cdot x^{2} + x (5 x) + x \cdot 6}{x^{2} - 4}$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (5 x)$ .

$\frac{x (x^{2} + 5 x) + x \cdot 6}{x^{2} - 4}$

Step 1.1.5

Factor $x$ out of $x (x^{2} + 5 x) + x \cdot 6$ .

$\frac{x (x^{2} + 5 x + 6)}{x^{2} - 4}$

Step 1.2

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 1.2.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.2.2

Write the factored form using these integers.

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

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 $b = 2$ .

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

Step 3

Cancel the common factor of $x + 2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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