Algebra Examples

$\frac{(x^{2} - x - 2) (x^{2} - 9)}{(x^{2} - 2 x - 3) (x^{2} + x - 6)}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 1) (x^{2} - 9)}{(x^{2} - 2 x - 3) (x^{2} + x - 6)}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\frac{(x - 2) (x + 1) (x^{2} - 3^{2})}{(x^{2} - 2 x - 3) (x^{2} + x - 6)}$

Step 1.3

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 $b = 3$ .

$\frac{(x - 2) (x + 1) (x + 3) (x - 3)}{(x^{2} - 2 x - 3) (x^{2} + x - 6)}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

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

$- 3, 1$

Step 2.1.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 1) (x + 3) (x - 3)}{(x - 3) (x + 1) (x^{2} + x - 6)}$

Step 2.2

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

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Step 2.2.1

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 $1$ .

$- 2, 3$

Step 2.2.2

Write the factored form using these integers.

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 1$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$\frac{(x + 3) (x - 3)}{(x - 3) (x + 3)}$

Step 3.3

Cancel the common factor of $x + 3$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x + 3) (x - 3)}{(x - 3) (x + 3)}$

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x - 3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

$1$