Precalculus Examples

$\frac{\frac{2 x^{2} - 3 x - 2}{x^{2} - 1}}{\frac{2 x^{2} + 5 x + 2}{x^{2} + x - 2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2} - 3 x - 2}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2

Factor by grouping.

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Step 2.1

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 2.1.1

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

$\frac{2 x^{2} - 3 (x) - 2}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2.1.2

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

$\frac{2 x^{2} + (1 - 4) x - 2}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2.1.3

Apply the distributive property.

$\frac{2 x^{2} + 1 x - 4 x - 2}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2.1.4

Multiply $x$ by $1$ .

$\frac{2 x^{2} + x - 4 x - 2}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + x) - 4 x - 2}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2.2.2

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

$\frac{x (2 x + 1) - 2 (2 x + 1)}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 2.3

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

$\frac{(2 x + 1) (x - 2)}{x^{2} - 1} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{(2 x + 1) (x - 2)}{x^{2} - 1^{2}} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 3.2

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 $b = 1$ .

$\frac{(2 x + 1) (x - 2)}{(x + 1) (x - 1)} \cdot \frac{x^{2} + x - 2}{2 x^{2} + 5 x + 2}$

Step 4

Factor $x^{2} + x - 2$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(2 x + 1) (x - 2)}{(x + 1) (x - 1)} \cdot \frac{(x - 1) (x + 2)}{2 x^{2} + 5 x + 2}$

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 $b = 5$ .

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Step 5.1.1

Factor $5$ out of $5 x$ .

$\frac{(2 x + 1) (x - 2)}{(x + 1) (x - 1)} \cdot \frac{(x - 1) (x + 2)}{2 x^{2} + 5 (x) + 2}$

Step 5.1.2

Rewrite $5$ as $1$ plus $4$

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Multiply $x$ by $1$ .

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

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.

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

Step 5.2.2

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

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

Step 5.3

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

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 1$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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

Step 9

Cancel the common factor of $x + 2$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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