Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor $x$ out of $x^{3} - 9 x$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor $x$ out of $- 9 x$ .

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

Step 1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

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

Step 1.2

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

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

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 $b = 3$ .

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Cancel the common factor of $x - 3$ .

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Step 3.2.1

Factor $x - 3$ out of $x (x + 3) (x - 3)$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 3$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Combine $\frac{x (x + 3)}{x + 3}$ and $x^{2}$ .

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

Step 4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.1

Move $x^{2}$ .

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

Step 4.2

Multiply $x^{2}$ by $x$ .

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Step 4.2.1

Raise $x$ to the power of $1$ .

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

Step 4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3

Add $2$ and $1$ .

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

Step 5

Cancel the common factor of $x + 3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $x^{3}$ by $1$ .

$x^{3}$