Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $2$ and $2 x^{3} - 2$ .

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Step 1.1.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{2 x^{3} - 2} - \frac{4}{3 x^{3} - 3}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

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

$\frac{2 (1)}{2 (x^{3}) - 2} - \frac{4}{3 x^{3} - 3}$

Step 1.1.2.2

Factor $2$ out of $- 2$ .

$\frac{2 \cdot 1}{2 (x^{3}) + 2 \cdot - 1} - \frac{4}{3 x^{3} - 3}$

Step 1.1.2.3

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

$\frac{2 (1)}{2 (x^{3} - 1)} - \frac{4}{3 x^{3} - 3}$

Step 1.1.2.4

Cancel the common factor.

$\frac{2 \cdot 1}{2 (x^{3} - 1)} - \frac{4}{3 x^{3} - 3}$

Step 1.1.2.5

Rewrite the expression.

$\frac{1}{x^{3} - 1} - \frac{4}{3 x^{3} - 3}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $1$ as $1^{3}$ .

$\frac{1}{x^{3} - 1^{3}} - \frac{4}{3 x^{3} - 3}$

Step 1.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$\frac{1}{(x - 1) (x^{2} + x \cdot 1 + 1^{2})} - \frac{4}{3 x^{3} - 3}$

Step 1.2.3

Simplify.

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Step 1.2.3.1

Multiply $x$ by $1$ .

$\frac{1}{(x - 1) (x^{2} + x + 1^{2})} - \frac{4}{3 x^{3} - 3}$

Step 1.2.3.2

One to any power is one.

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 x^{3} - 3}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

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

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Step 1.3.1.1

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

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 (x^{3}) - 3}$

Step 1.3.1.2

Factor $3$ out of $- 3$ .

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 (x^{3}) + 3 (- 1)}$

Step 1.3.1.3

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

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 (x^{3} - 1)}$

Step 1.3.2

Rewrite $1$ as $1^{3}$ .

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 (x^{3} - 1^{3})}$

Step 1.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 ((x - 1) (x^{2} + x \cdot 1 + 1^{2}))}$

Step 1.3.4

Simplify.

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Step 1.3.4.1

Multiply $x$ by $1$ .

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 ((x - 1) (x^{2} + x + 1^{2}))}$

Step 1.3.4.2

One to any power is one.

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 ((x - 1) (x^{2} + x + 1))}$

$\frac{1}{(x - 1) (x^{2} + x + 1)} - \frac{4}{3 (x - 1) (x^{2} + x + 1)}$

Step 2

To write $\frac{1}{(x - 1) (x^{2} + x + 1)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{(x - 1) (x^{2} + x + 1)} \cdot \frac{3}{3} - \frac{4}{3 (x - 1) (x^{2} + x + 1)}$

Step 3

Write each expression with a common denominator of $(x - 1) (x^{2} + x + 1) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\frac{3}{(x - 1) (x^{2} + x + 1) \cdot 3} - \frac{4}{3 (x - 1) (x^{2} + x + 1)}$

Step 3.2

Reorder the factors of $(x - 1) (x^{2} + x + 1) \cdot 3$ .

$\frac{3}{3 (x - 1) (x^{2} + x + 1)} - \frac{4}{3 (x - 1) (x^{2} + x + 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 - 4}{3 (x - 1) (x^{2} + x + 1)}$

Step 5

Simplify the expression.

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Step 5.1

Subtract $4$ from $3$ .

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

Step 5.2

Move the negative in front of the fraction.

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