Calculus Examples

$\frac{x + 2}{x^{3} + 8}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

$\frac{x + 2}{x^{3} + 2^{3}}$

Step 1.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $- 1$ .

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

Step 1.3.2

Raise $2$ to the power of $2$ .

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

Step 2

Cancel the common factor of $x + 2$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

$\frac{1}{x^{2} - 2 x + 4}$