Algebra Examples

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

Step 1

Factor $6 x^{2}$ out of $6 x^{3} + 6 x^{2}$ .

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Step 1.1

Factor $6 x^{2}$ out of $6 x^{3}$ .

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

Step 1.2

Factor $6 x^{2}$ out of $6 x^{2}$ .

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

Step 1.3

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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

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 $b = 1$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.2.1

Factor $x$ out of $3 x^{2} (x + 1)$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\frac{3 x}{x - 1}$