Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Reorder terms.

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $- x + 2$ and $x - 2$ .

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Step 3.1.1

Factor $- 1$ out of $- x$ .

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

Step 3.1.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 3.1.3

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

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

Step 3.1.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 3.1.5

Cancel the common factor.

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

Step 3.1.6

Rewrite the expression.

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

Step 3.2

Move the negative in front of the fraction.

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