Algebra Examples

$\frac{2 x^{3} + 4 x^{2} + 2 x}{6 x^{3} - 6 x}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Factor $2 x$ out of $2 x^{3} + 4 x^{2} + 2 x$ .

Tap for more steps...

Step 1.1.1

Factor $2 x$ out of $2 x^{3}$ .

$\frac{2 x (x^{2}) + 4 x^{2} + 2 x}{6 x^{3} - 6 x}$

Step 1.1.2

Factor $2 x$ out of $4 x^{2}$ .

$\frac{2 x (x^{2}) + 2 x (2 x) + 2 x}{6 x^{3} - 6 x}$

Step 1.1.3

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

$\frac{2 x (x^{2}) + 2 x (2 x) + 2 x (1)}{6 x^{3} - 6 x}$

Step 1.1.4

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

$\frac{2 x (x^{2} + 2 x) + 2 x (1)}{6 x^{3} - 6 x}$

Step 1.1.5

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

$\frac{2 x (x^{2} + 2 x + 1)}{6 x^{3} - 6 x}$

Step 1.2

Factor using the perfect square rule.

Tap for more steps...

Step 1.2.1

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

$\frac{2 x (x^{2} + 2 x + 1^{2})}{6 x^{3} - 6 x}$

Step 1.2.2

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 1.2.3

Rewrite the polynomial.

$\frac{2 x (x^{2} + 2 \cdot x \cdot 1 + 1^{2})}{6 x^{3} - 6 x}$

Step 1.2.4

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

$\frac{2 x {(x + 1)}^{2}}{6 x^{3} - 6 x}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Factor $6 x$ out of $6 x^{3} - 6 x$ .

Tap for more steps...

Step 2.1.1

Factor $6 x$ out of $6 x^{3}$ .

$\frac{2 x {(x + 1)}^{2}}{6 x (x^{2}) - 6 x}$

Step 2.1.2

Factor $6 x$ out of $- 6 x$ .

$\frac{2 x {(x + 1)}^{2}}{6 x (x^{2}) + 6 x (- 1)}$

Step 2.1.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 1)$ .

$\frac{2 x {(x + 1)}^{2}}{6 x (x^{2} - 1)}$

Step 2.2

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

$\frac{2 x {(x + 1)}^{2}}{6 x (x^{2} - 1^{2})}$

Step 2.3

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 $b = 1$ .

$\frac{2 x {(x + 1)}^{2}}{6 x (x + 1) (x - 1)}$

Step 3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.1

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 3.1.1

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

$\frac{2 (x {(x + 1)}^{2})}{6 x (x + 1) (x - 1)}$

Step 3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.1.2.1

Factor $2$ out of $6 x (x + 1) (x - 1)$ .

$\frac{2 (x {(x + 1)}^{2})}{2 (3 x (x + 1) (x - 1))}$

Step 3.1.2.2

Cancel the common factor.

$\frac{2 (x {(x + 1)}^{2})}{2 (3 x (x + 1) (x - 1))}$

Step 3.1.2.3

Rewrite the expression.

$\frac{x {(x + 1)}^{2}}{3 x (x + 1) (x - 1)}$

Step 3.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

$\frac{x {(x + 1)}^{2}}{3 x (x + 1) (x - 1)}$

Step 3.2.2

Rewrite the expression.

$\frac{{(x + 1)}^{2}}{(3 (x + 1)) (x - 1)}$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$\frac{(x + 1) (x + 1)}{(3 (x + 1)) (x - 1)}$

Step 3.3.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1

Factor $x + 1$ out of $(3 (x + 1)) (x - 1)$ .

$\frac{(x + 1) (x + 1)}{(x + 1) ((3) (x - 1))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{(x + 1) (x + 1)}{(x + 1) ((3) (x - 1))}$

Step 3.3.2.3

Rewrite the expression.

$\frac{x + 1}{(3) (x - 1)}$

$\frac{x + 1}{3 (x - 1)}$