Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Factor $3 x$ out of $3 x$ .

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

Step 1.1.3

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Cancel the common factor of $3$ and $6$ .

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $3$ out of $6 x (x - 1)$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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 $b = 1$ .

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

Step 2.3

Combine exponents.

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Step 2.3.1

Raise $x + 1$ to the power of $1$ .

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

Step 2.3.2

Raise $x + 1$ to the power of $1$ .

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

Step 2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 3

Cancel the common factor of $x - 1$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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