Basic Math Examples

$\frac{x^{3} - 8}{2 - x}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

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

Step 1.2

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 $b = 2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Move $2$ to the left of $x$ .

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

Step 1.3.2

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

$\frac{(x - 2) (x^{2} + 2 x + 4)}{2 - x}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x - 2$ and $2 - x$ .

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Step 2.1.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) - 2) (x^{2} + 2 x + 4)}{2 - x}$

Step 2.1.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- 1 (- x) - 1 (2)) (x^{2} + 2 x + 4)}{2 - x}$

Step 2.1.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

$\frac{- 1 (- x + 2) (x^{2} + 2 x + 4)}{2 - x}$

Step 2.1.4

Reorder terms.

$\frac{- 1 (- x + 2) (x^{2} + 2 x + 4)}{- x + 2}$

Step 2.1.5

Cancel the common factor.

$\frac{- 1 (- x + 2) (x^{2} + 2 x + 4)}{- x + 2}$

Step 2.1.6

Divide $- 1 (x^{2} + 2 x + 4)$ by $1$ .

$- 1 (x^{2} + 2 x + 4)$

Step 2.2

Rewrite $- 1 (x^{2} + 2 x + 4)$ as $- (x^{2} + 2 x + 4)$ .

$- (x^{2} + 2 x + 4)$

Step 2.3

Apply the distributive property.

$- x^{2} - (2 x) - 1 \cdot 4$

Step 3

Simplify.

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Step 3.1

Multiply $2$ by $- 1$ .

$- x^{2} - 2 x - 1 \cdot 4$

Step 3.2

Multiply $- 1$ by $4$ .

$- x^{2} - 2 x - 4$