Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

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Step 2.1

Raise $x$ to the power of $1$ .

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

Step 2.2

Factor $x$ out of $x^{1}$ .

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

Step 2.3

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

$\frac{x \cdot 1 + x (- x)}{x^{3} - 1} \cdot \frac{x^{2} + x + 1}{x}$

Step 2.4

Factor $x$ out of $x \cdot 1 + x (- x)$ .

$\frac{x (1 - x)}{x^{3} - 1} \cdot \frac{x^{2} + x + 1}{x}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{x (1 - x)}{x^{3} - 1^{3}} \cdot \frac{x^{2} + x + 1}{x}$

Step 3.2

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 $b = 1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Multiply $x$ by $1$ .

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

Step 3.3.2

One to any power is one.

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x^{2} + x + 1$ .

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Step 4.2.1

Factor $x^{2} + x + 1$ out of $(x - 1) (x^{2} + x + 1)$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{1 - x}{x - 1}$

Step 4.3

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 4.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1) - x}{x - 1}$

Step 4.3.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- 1) - (x)}{x - 1}$

Step 4.3.3

Factor $- 1$ out of $- 1 (- 1) - (x)$ .

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

Step 4.3.4

Reorder terms.

$\frac{- 1 (x - 1)}{x - 1}$

Step 4.3.5

Cancel the common factor.

$\frac{- 1 (x - 1)}{x - 1}$

Step 4.3.6

Divide $- 1$ by $1$ .

$- 1$