Trigonometry Examples

$f (x) = \frac{x^{3} - 1}{x - 1}$

Step 1

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

$f (x) = \frac{x^{3} - 1^{3}}{x - 1}$

Step 2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$f (x) = \frac{(x - 1) (x^{2} + x \cdot 1 + 1^{2})}{x - 1}$

Step 3

Simplify.

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Step 3.1

Multiply $x$ by $1$ .

$f (x) = \frac{(x - 1) (x^{2} + x + 1^{2})}{x - 1}$

Step 3.2

One to any power is one.

$f (x) = \frac{(x - 1) (x^{2} + x + 1)}{x - 1}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Reduce the expression $\frac{(x - 1) (x^{2} + x + 1)}{x - 1}$ by cancelling the common factors.

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Step 4.1.1

Cancel the common factor.

$f (x) = \frac{(x - 1) (x^{2} + x + 1)}{x - 1}$

Step 4.1.2

Rewrite the expression.

$f (x) = \frac{x^{2} + x + 1}{1}$

Step 4.2

Divide $x^{2} + x + 1$ by $1$ .

$f (x) = x^{2} + x + 1$