Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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 $b = 1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 6

Factor $2 x$ out of $2 x^{2} + 2 x$ .

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Step 6.1

Factor $2 x$ out of $2 x^{2}$ .

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

Step 6.2

Factor $2 x$ out of $2 x$ .

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

Step 6.3

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

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

Step 7

Cancel the common factor of $x + 1$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Multiply $\frac{1}{x - 1}$ by $\frac{1}{2 x}$ .

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

Step 9

Move $2$ to the left of $x - 1$ .

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

Step 10

Reorder factors in $\frac{1}{2 (x - 1) x}$ .

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