Algebra Examples

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

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

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

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

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

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

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 2.2

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

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

Step 2.3

Rewrite the polynomial.

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

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

Step 2.4

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = x

$a = x$ and

b = 1

$b = 1$ .

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

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

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

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

Step 3

Cancel the common factor of

x - 1

$x - 1$ and

{(x - 1)}^{2}

${(x - 1)}^{2}$ .

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Step 3.1

Factor

x - 1

$x - 1$ out of

(x + 1) (x - 1)

$(x + 1) (x - 1)$ .

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor

x - 1

$x - 1$ out of

{(x - 1)}^{2}

${(x - 1)}^{2}$ .

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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