Precalculus Examples

$\frac{2 x^{3} - 28 x^{2} - 102 x}{18 x - 2 x^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x$ out of $2 x^{3} - 28 x^{2} - 102 x$ .

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Step 1.1.1

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

$\frac{2 x (x^{2}) - 28 x^{2} - 102 x}{18 x - 2 x^{3}}$

Step 1.1.2

Factor $2 x$ out of $- 28 x^{2}$ .

$\frac{2 x (x^{2}) + 2 x (- 14 x) - 102 x}{18 x - 2 x^{3}}$

Step 1.1.3

Factor $2 x$ out of $- 102 x$ .

$\frac{2 x (x^{2}) + 2 x (- 14 x) + 2 x (- 51)}{18 x - 2 x^{3}}$

Step 1.1.4

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 14 x)$ .

$\frac{2 x (x^{2} - 14 x) + 2 x (- 51)}{18 x - 2 x^{3}}$

Step 1.1.5

Factor $2 x$ out of $2 x (x^{2} - 14 x) + 2 x (- 51)$ .

$\frac{2 x (x^{2} - 14 x - 51)}{18 x - 2 x^{3}}$

Step 1.2

Factor $x^{2} - 14 x - 51$ 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 $- 51$ and whose sum is $- 14$ .

$- 17, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{2 x ((x - 17) (x + 3))}{18 x - 2 x^{3}}$

$\frac{2 x (x - 17) (x + 3)}{18 x - 2 x^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2 x$ out of $18 x - 2 x^{3}$ .

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Step 2.1.1

Factor $2 x$ out of $18 x$ .

$\frac{2 x (x - 17) (x + 3)}{2 x (9) - 2 x^{3}}$

Step 2.1.2

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

$\frac{2 x (x - 17) (x + 3)}{2 x (9) + 2 x (- x^{2})}$

Step 2.1.3

Factor $2 x$ out of $2 x (9) + 2 x (- x^{2})$ .

$\frac{2 x (x - 17) (x + 3)}{2 x (9 - x^{2})}$

Step 2.2

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

$\frac{2 x (x - 17) (x + 3)}{2 x (3^{2} - x^{2})}$

Step 2.3

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

$\frac{2 x (x - 17) (x + 3)}{2 x (3 + x) (3 - x)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Cancel the common factor.

$\frac{2 x (x - 17) (x + 3)}{2 x (3 + x) (3 - x)}$

Step 3.1.2

Rewrite the expression.

$\frac{(x (x - 17)) (x + 3)}{(x (3 + x)) (3 - x)}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x (x - 17) (x + 3)}{x (3 + x) (3 - x)}$

Step 3.2.2

Rewrite the expression.

$\frac{(x - 17) (x + 3)}{(3 + x) (3 - x)}$

Step 3.3

Cancel the common factor of $x + 3$ and $3 + x$ .

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Step 3.3.1

Reorder terms.

$\frac{(x - 17) (x + 3)}{(x + 3) (3 - x)}$

Step 3.3.2

Cancel the common factor.

$\frac{(x - 17) (x + 3)}{(x + 3) (3 - x)}$

Step 3.3.3

Rewrite the expression.

$\frac{x - 17}{3 - x}$