Algebra Examples

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

Step 1

Factor $5$ out of $5 x + 5$ .

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Step 1.1

Factor $5$ out of $5 x$ .

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

Step 1.2

Factor $5$ out of $5$ .

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

Step 1.3

Factor $5$ out of $5 (x) + 5 (1)$ .

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

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.

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

Step 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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 $b = 1$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 1$ .

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Step 4.1.1

Factor $x + 1$ out of $5 (x + 1)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$5 \cdot \frac{x - 2}{x - 1}$

Step 4.3

Combine $5$ and $\frac{x - 2}{x - 1}$ .

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