Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

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

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

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.

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

Step 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

Divide $x + 1$ by $1$ .

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

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

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 $b = 1$ .

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

Step 5

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $x + 1$ by $1$ .

$(x + 1) (x + 1)$

Step 6

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Tap for more steps...

Step 6.1

Apply the distributive property.

$x (x + 1) + 1 (x + 1)$

Step 6.2

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 (x + 1)$

Step 6.3

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Step 7

Simplify and combine like terms.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply $x$ by $x$ .

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

Step 7.1.2

Multiply $x$ by $1$ .

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

Step 7.1.3

Multiply $x$ by $1$ .

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

Step 7.1.4

Multiply $1$ by $1$ .

$x^{2} + x + x + 1$

Step 7.2

Add $x$ and $x$ .

$x^{2} + 2 x + 1$