Precalculus Examples

$\frac{z + 1}{z^{3} + 1}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{3}$ .

$\frac{z + 1}{z^{3} + 1^{3}}$

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 $a = z$ and $b = 1$ .

$\frac{z + 1}{(z + 1) (z^{2} - z \cdot 1 + 1^{2})}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 1$ by $1$ .

$\frac{z + 1}{(z + 1) (z^{2} - z + 1^{2})}$

Step 1.3.2

One to any power is one.

$\frac{z + 1}{(z + 1) (z^{2} - z + 1)}$

Step 2

Cancel the common factor of $z + 1$ .

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Step 2.1

Cancel the common factor.

$\frac{z + 1}{(z + 1) (z^{2} - z + 1)}$

Step 2.2

Rewrite the expression.

$\frac{1}{z^{2} - z + 1}$