Find the Quotient of the Two Functions P(x)=x^4-2x^2-8 , d(x)=x+2

$P (x) = x^{4} - 2 x^{2} - 8$ , $d (x) = x + 2$

Step 1

Replace the function designators with the actual functions in $\frac{P (x)}{d (x)}$ .

$\frac{x^{4} - 2 x^{2} - 8}{x + 2}$

Step 2

Simplify.

Simplify the numerator.

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

$\frac{{(x^{2})}^{2} - 2 x^{2} - 8}{x + 2}$

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

$\frac{u^{2} - 2 u - 8}{x + 2}$

Factor $u^{2} - 2 u - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

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

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

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

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

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

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 $b = 2$ .

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

Multiply $- 2$ by $2$ .

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

Find the Quotient of the Two Functions P(x)=x^4-2x^2-8 , d(x)=x+2

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