Simplify 1/(x^3)-x

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

Step 1

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

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

Step 2

Simplify terms.

Combine $- x$ and $\frac{x^{3}}{x^{3}}$ .

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

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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

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

Rewrite $x \cdot x^{3}$ as ${(x^{2})}^{2}$ .

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

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

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

Simplify.

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

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

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

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

Simplify 1/(x^3)-x

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