Precalculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

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 $b = 1$ .

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

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 3.1.1

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

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

Step 3.1.2

Rewrite $- 1$ as $1$ plus $- 2$

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

Multiply $x$ by $1$ .

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

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 4

Cancel the common factor of $x - 1$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Factor by grouping.

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Step 5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 5.1.1

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

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

Step 5.1.2

Rewrite $- 3$ as $1$ plus $- 4$

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Multiply $x$ by $1$ .

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

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

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

Step 5.2.2

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

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

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

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

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

Step 6.2

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 $b = 2$ .

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $2 x + 1$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x - 2$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

Multiply $x + 1$ by $\frac{1}{x + 2}$ .

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