Pre-Algebra Examples

$\frac{x^{4} - 16}{x^{2} + 4}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(x^{2})}^{2} - 16}{x^{2} + 4}$

Step 1.2

Rewrite $16$ as $4^{2}$ .

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

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 $a = x^{2}$ and $b = 4$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.4.2

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 $b = 2$ .

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

Step 2

Cancel the common factor of $x^{2} + 4$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Divide $(x + 2) (x - 2)$ by $1$ .

$(x + 2) (x - 2)$

Step 3

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$x (x - 2) + 2 (x - 2)$

Step 3.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 + 2 (x - 2)$

Step 3.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$

Step 4

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 4.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$x \cdot x - 2 x + 2 x + 2 \cdot - 2$

Step 4.2

Add $- 2 x$ and $2 x$ .

$x \cdot x + 0 + 2 \cdot - 2$

Step 4.3

Add $x \cdot x$ and $0$ .

$x \cdot x + 2 \cdot - 2$

Step 5

Simplify each term.

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Step 5.1

Multiply $x$ by $x$ .

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

Step 5.2

Multiply $2$ by $- 2$ .

$x^{2} - 4$