Simplify ((x-1)/(x+1))÷((x^2-2x+1)/(x^2-x-2))

$\frac{x - 1}{x + 1} \div \frac{x^{2} - 2 x + 1}{x^{2} - x - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

$- 2, 1$

Write the factored form using these integers.

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

Step 3

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

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

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

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 1$ .

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $(x - 2) (x + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 2}{x - 1}$

Simplify ((x-1)/(x+1))÷((x^2-2x+1)/(x^2-x-2))

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