Divide (x^4-12x^2-64)/(x+4)

$\frac{x^{4} - 12 x^{2} - 64}{x + 4}$

Step 1

Simplify the numerator.

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

$\frac{{(x^{2})}^{2} - 12 x^{2} - 64}{x + 4}$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$\frac{u^{2} - 12 u - 64}{x + 4}$

Factor $u^{2} - 12 u - 64$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 64$ and whose sum is $- 12$ .

$- 16, 4$

Write the factored form using these integers.

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

Replace all occurrences of $u$ with $x^{2}$ .

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

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

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

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 $b = 4$ .

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

Step 2

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

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

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

Step 3

Expand $(x - 4) (x^{2} + 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify each term.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $2$ .

$x^{3} + x \cdot 4 - 4 x^{2} - 4 \cdot 4$

Move $4$ to the left of $x$ .

$x^{3} + 4 \cdot x - 4 x^{2} - 4 \cdot 4$

Multiply $- 4$ by $4$ .

$x^{3} + 4 x - 4 x^{2} - 16$

Divide (x^4-12x^2-64)/(x+4)

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