Simplify ((x-1)^2(x^2-1))/((x-1)^2(x^2-4x-5))

$\frac{{(x - 1)}^{2} (x^{2} - 1)}{{(x - 1)}^{2} (x^{2} - 4 x - 5)}$

Step 1

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

Cancel the common factor.

$\frac{{(x - 1)}^{2} (x^{2} - 1)}{{(x - 1)}^{2} (x^{2} - 4 x - 5)}$

Rewrite the expression.

$\frac{x^{2} - 1}{x^{2} - 4 x - 5}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 1^{2}}{x^{2} - 4 x - 5}$

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 $b = 1$ .

$\frac{(x + 1) (x - 1)}{x^{2} - 4 x - 5}$

Step 3

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Write the factored form using these integers.

$\frac{(x + 1) (x - 1)}{(x - 5) (x + 1)}$

Step 4

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{(x - 5) (x + 1)}$

Rewrite the expression.

$\frac{x - 1}{x - 5}$

Simplify ((x-1)^2(x^2-1))/((x-1)^2(x^2-4x-5))

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