Precalculus Examples

$\frac{x^{2} - x - 6}{x^{2} + 2 x} \cdot \frac{x^{3} + x^{2}}{x^{2} - 2 x - 3}$

Step 1

Factor $x^{2} - x - 6$ using the AC method.

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Step 1.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify with factoring out.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $2 x$ .

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

Step 2.1.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply by $1$ .

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

Step 2.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 1$ .

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

Step 3

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 3.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $x - 3$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $x + 2$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

$x$