Simplify ((x^2-16x+64)(x+2))/((x^2-64)(x^2-6x-16))

$\frac{(x^{2} - 16 x + 64) (x + 2)}{(x^{2} - 64) (x^{2} - 6 x - 16)}$

Step 1

Factor using the perfect square rule.

Rewrite $64$ as $8^{2}$ .

$\frac{(x^{2} - 16 x + 8^{2}) (x + 2)}{(x^{2} - 64) (x^{2} - 6 x - 16)}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Rewrite the polynomial.

$\frac{(x^{2} - 2 \cdot x \cdot 8 + 8^{2}) (x + 2)}{(x^{2} - 64) (x^{2} - 6 x - 16)}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 8$ .

$\frac{{(x - 8)}^{2} (x + 2)}{(x^{2} - 64) (x^{2} - 6 x - 16)}$

Step 2

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$\frac{{(x - 8)}^{2} (x + 2)}{(x^{2} - 8^{2}) (x^{2} - 6 x - 16)}$

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 $b = 8$ .

$\frac{{(x - 8)}^{2} (x + 2)}{(x + 8) (x - 8) (x^{2} - 6 x - 16)}$

Factor $x^{2} - 6 x - 16$ 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 $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Write the factored form using these integers.

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

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

Combine exponents.

Raise $x - 8$ to the power of $1$ .

$\frac{{(x - 8)}^{2} (x + 2)}{(x + 8) ({(x - 8)}^{1} (x - 8)) (x + 2)}$

Raise $x - 8$ to the power of $1$ .

$\frac{{(x - 8)}^{2} (x + 2)}{(x + 8) ({(x - 8)}^{1} {(x - 8)}^{1}) (x + 2)}$

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

$\frac{{(x - 8)}^{2} (x + 2)}{(x + 8) {(x - 8)}^{1 + 1} (x + 2)}$

Add $1$ and $1$ .

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of ${(x - 8)}^{2}$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x + 8}$

Simplify ((x^2-16x+64)(x+2))/((x^2-64)(x^2-6x-16))

Related Questions

Support LCMGCF.com by disabling your adblocker.