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

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

Step 1

Simplify the numerator.

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

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

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 $b = 1$ .

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

Step 2

Factor using the perfect square rule.

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

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

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.

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

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

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

Step 3

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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