Basic Math Examples

$\frac{x^{3} + 2 x^{2} - 15 x}{x^{3} - 27}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x$ out of $x^{3} + 2 x^{2} - 15 x$ .

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Step 1.1.1

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

$\frac{x \cdot x^{2} + 2 x^{2} - 15 x}{x^{3} - 27}$

Step 1.1.2

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

$\frac{x \cdot x^{2} + x (2 x) - 15 x}{x^{3} - 27}$

Step 1.1.3

Factor $x$ out of $- 15 x$ .

$\frac{x \cdot x^{2} + x (2 x) + x \cdot - 15}{x^{3} - 27}$

Step 1.1.4

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

$\frac{x (x^{2} + 2 x) + x \cdot - 15}{x^{3} - 27}$

Step 1.1.5

Factor $x$ out of $x (x^{2} + 2 x) + x \cdot - 15$ .

$\frac{x (x^{2} + 2 x - 15)}{x^{3} - 27}$

Step 1.2

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 1.2.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{x ((x - 3) (x + 5))}{x^{3} - 27}$

$\frac{x (x - 3) (x + 5)}{x^{3} - 27}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $27$ as $3^{3}$ .

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

Step 2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 3$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Move $3$ to the left of $x$ .

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

Step 2.3.2

Raise $3$ to the power of $2$ .

$\frac{x (x - 3) (x + 5)}{(x - 3) (x^{2} + 3 x + 9)}$

Step 3

Cancel the common factor of $x - 3$ .

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Step 3.1

Cancel the common factor.

$\frac{x (x - 3) (x + 5)}{(x - 3) (x^{2} + 3 x + 9)}$

Step 3.2

Rewrite the expression.

$\frac{x (x + 5)}{x^{2} + 3 x + 9}$