Divide (a^(2n)-a^n-6)/(a^n+8)

$\frac{a^{2 n} - a^{n} - 6}{a^{n} + 8}$

Step 1

Simplify the numerator.

Rewrite $a^{2 n}$ as ${(a^{n})}^{2}$ .

$\frac{{(a^{n})}^{2} - a^{n} - 6}{a^{n} + 8}$

Let $u = a^{n}$ . Substitute $u$ for all occurrences of $a^{n}$ .

$\frac{u^{2} - u - 6}{a^{n} + 8}$

Factor $u^{2} - u - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

$\frac{(u - 3) (u + 2)}{a^{n} + 8}$

Replace all occurrences of $u$ with $a^{n}$ .

$\frac{(a^{n} - 3) (a^{n} + 2)}{a^{n} + 8}$

Step 2

Expand $(a^{n} - 3) (a^{n} + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{a^{n} (a^{n} + 2) - 3 (a^{n} + 2)}{a^{n} + 8}$

Apply the distributive property.

$\frac{a^{n} a^{n} + a^{n} \cdot 2 - 3 (a^{n} + 2)}{a^{n} + 8}$

Apply the distributive property.

$\frac{a^{n} a^{n} + a^{n} \cdot 2 - 3 a^{n} - 3 \cdot 2}{a^{n} + 8}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $a^{n}$ by $a^{n}$ by adding the exponents.

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

$\frac{a^{n + n} + a^{n} \cdot 2 - 3 a^{n} - 3 \cdot 2}{a^{n} + 8}$

Add $n$ and $n$ .

$\frac{a^{2 n} + a^{n} \cdot 2 - 3 a^{n} - 3 \cdot 2}{a^{n} + 8}$

Move $2$ to the left of $a^{n}$ .

$\frac{a^{2 n} + 2 \cdot a^{n} - 3 a^{n} - 3 \cdot 2}{a^{n} + 8}$

Multiply $- 3$ by $2$ .

$\frac{a^{2 n} + 2 a^{n} - 3 a^{n} - 6}{a^{n} + 8}$

Subtract $3 a^{n}$ from $2 a^{n}$ .

$\frac{a^{2 n} - a^{n} - 6}{a^{n} + 8}$

Step 4

Split the fraction $\frac{a^{2 n} - a^{n} - 6}{a^{n} + 8}$ into two fractions.

$\frac{a^{2 n} - a^{n}}{a^{n} + 8} + \frac{- 6}{a^{n} + 8}$

Step 5

Split the fraction $\frac{a^{2 n} - a^{n}}{a^{n} + 8}$ into two fractions.

$\frac{a^{2 n}}{a^{n} + 8} + \frac{- a^{n}}{a^{n} + 8} + \frac{- 6}{a^{n} + 8}$

Step 6

Move the negative in front of the fraction.

$\frac{a^{2 n}}{a^{n} + 8} - \frac{a^{n}}{a^{n} + 8} + \frac{- 6}{a^{n} + 8}$

Step 7

Move the negative in front of the fraction.

$\frac{a^{2 n}}{a^{n} + 8} - \frac{a^{n}}{a^{n} + 8} - \frac{6}{a^{n} + 8}$

Divide (a^(2n)-a^n-6)/(a^n+8)

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