Algebra Examples

$(x^{3} + 3 x^{2} - 4 x - 12) \div (x^{2} + 5 x + 6)$

Step 1

Rewrite the division as a fraction.

$\frac{x^{3} + 3 x^{2} - 4 x - 12}{x^{2} + 5 x + 6}$

Step 2

Simplify the numerator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$\frac{(x^{3} + 3 x^{2}) - 4 x - 12}{x^{2} + 5 x + 6}$

Step 2.1.2

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

$\frac{x^{2} (x + 3) - 4 (x + 3)}{x^{2} + 5 x + 6}$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

$\frac{(x + 3) (x^{2} - 4)}{x^{2} + 5 x + 6}$

Step 2.3

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

$\frac{(x + 3) (x^{2} - 2^{2})}{x^{2} + 5 x + 6}$

Step 2.4

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 $b = 2$ .

$\frac{(x + 3) (x + 2) (x - 2)}{x^{2} + 5 x + 6}$

Step 3

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Step 3.2

Write the factored form using these integers.

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

Step 4

Cancel the common factor of $x + 3$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Cancel the common factor of $x + 2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $x - 2$ by $1$ .

$x - 2$