Precalculus Examples

$\frac{x^{3} - 9 x}{3 - x}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x$ out of $x^{3} - 9 x$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor $x$ out of $- 9 x$ .

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

Step 1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

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

Step 1.2

Rewrite $9$ as $3^{2}$ .

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

Step 1.3

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 $b = 3$ .

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Factor $- 1$ out of $x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Reorder terms.

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

Step 2.1.5

Cancel the common factor.

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

Step 2.1.6

Divide $x (x + 3) \cdot (- 1)$ by $1$ .

$x (x + 3) \cdot (- 1)$

Step 2.2

Simplify by multiplying through.

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Step 2.2.1

Apply the distributive property.

$(x \cdot x + x \cdot 3) \cdot - 1$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2.2

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

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

Step 2.2.3

Apply the distributive property.

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

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 2.2.4.2

Multiply $- 1$ by $3$ .

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

Step 2.3

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

$- x^{2} - 3 x$