Simplify (x^2)/(x^2-16)+(8(x-2))/(16-x^2)

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

Step 1

Simplify terms.

Simplify each term.

Simplify the denominator.

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

$\frac{x^{2}}{x^{2} - 4^{2}} + \frac{8 (x - 2)}{16 - x^{2}}$

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 $b = 4$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{16 - x^{2}}$

Simplify the denominator.

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

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

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

Simplify terms.

Reorder terms.

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

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{(x + 4) (- 1 (- 4) - x)}$

Factor $- 1$ out of $- x$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{(x + 4) (- 1 (- 4) - (x))}$

Factor $- 1$ out of $- 1 (- 4) - (x)$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{(x + 4) (- 1 (- 4 + x))}$

Simplify the expression.

Move a negative from the denominator of $\frac{8 (x - 2)}{(x + 4) (- 1 (- 4 + x))}$ to the numerator.

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

Reorder terms.

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

Multiply $- 1$ by $8$ .

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

Apply the distributive property.

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

Multiply $- 8$ by $- 2$ .

$\frac{x^{2} - 8 x + 16}{(x + 4) (x - 4)}$

Factor using the perfect square rule.

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

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

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 3

Cancel the common factor of ${(x - 4)}^{2}$ and $x - 4$ .

Factor $x - 4$ out of ${(x - 4)}^{2}$ .

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

Cancel the common factors.

Factor $x - 4$ out of $(x + 4) (x - 4)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 4}{x + 4}$

Simplify (x^2)/(x^2-16)+(8(x-2))/(16-x^2)

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