Pre-Algebra Examples

$\frac{3 x - x^{2}}{x^{3} - 27} \div \frac{x}{x^{2} + 3 x + 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x - x^{2}}{x^{3} - 27} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 2

Factor $x$ out of $3 x - x^{2}$ .

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Step 2.1

Factor $x$ out of $3 x$ .

$\frac{x \cdot 3 - x^{2}}{x^{3} - 27} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 2.2

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

$\frac{x \cdot 3 + x (- x)}{x^{3} - 27} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 2.3

Factor $x$ out of $x \cdot 3 + x (- x)$ .

$\frac{x (3 - x)}{x^{3} - 27} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $27$ as $3^{3}$ .

$\frac{x (3 - x)}{x^{3} - 3^{3}} \cdot \frac{x^{2} + 3 x + 9}{x}$

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 $b = 3$ .

$\frac{x (3 - x)}{(x - 3) (x^{2} + x \cdot 3 + 3^{2})} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 3.3

Simplify.

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Step 3.3.1

Move $3$ to the left of $x$ .

$\frac{x (3 - x)}{(x - 3) (x^{2} + 3 \cdot x + 3^{2})} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 3.3.2

Raise $3$ to the power of $2$ .

$\frac{x (3 - x)}{(x - 3) (x^{2} + 3 x + 9)} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 4

Cancel the common factor of $x$ .

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Step 4.1

Cancel the common factor.

$\frac{x (3 - x)}{(x - 3) (x^{2} + 3 x + 9)} \cdot \frac{x^{2} + 3 x + 9}{x}$

Step 4.2

Rewrite the expression.

$\frac{3 - x}{(x - 3) (x^{2} + 3 x + 9)} (x^{2} + 3 x + 9)$

Step 5

Cancel the common factor of $x^{2} + 3 x + 9$ .

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Step 5.1

Factor $x^{2} + 3 x + 9$ out of $(x - 3) (x^{2} + 3 x + 9)$ .

$\frac{3 - x}{(x^{2} + 3 x + 9) (x - 3)} (x^{2} + 3 x + 9)$

Step 5.2

Cancel the common factor.

$\frac{3 - x}{(x^{2} + 3 x + 9) (x - 3)} (x^{2} + 3 x + 9)$

Step 5.3

Rewrite the expression.

$\frac{3 - x}{x - 3}$

Step 6

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 6.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 1 (- 3) - x}{x - 3}$

Step 6.2

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

$\frac{- 1 (- 3) - (x)}{x - 3}$

Step 6.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

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

Step 6.4

Reorder terms.

$\frac{- 1 (x - 3)}{x - 3}$

Step 6.5

Cancel the common factor.

$\frac{- 1 (x - 3)}{x - 3}$

Step 6.6

Divide $- 1$ by $1$ .

$- 1$