Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

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.

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

Step 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

Cancel the common factor of $x + 1$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Cancel the common factor of $x - 1$ and $1 - x$ .

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Step 5.1

Factor $- 1$ out of $x$ .

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

Step 5.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 5.3

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

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

Step 5.4

Reorder terms.

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

Step 5.5

Cancel the common factor.

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

Step 5.6

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

$- 1 (x + 1)$

Step 6

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

$- (x + 1)$

Step 7

Apply the distributive property.

$- x - 1 \cdot 1$

Step 8

Multiply $- 1$ by $1$ .

$- x - 1$