Algebra Examples

$\frac{x^{3} - 1}{x^{2} - 1}$

Step 1

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

$\frac{x^{3} - 1^{3}}{x^{2} - 1}$

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 $b = 1$ .

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

Step 3

Simplify.

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Step 3.1

Multiply $x$ by $1$ .

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

Step 3.2

One to any power is one.

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

Step 4

Rewrite $1$ as $1^{2}$ .

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

Step 5

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 $b = 1$ .

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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