Algebra Examples

$\frac{5 x^{2} + 5 x - 100}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $5$ out of $5 x^{2} + 5 x - 100$ .

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Step 1.1.1

Factor $5$ out of $5 x^{2}$ .

$\frac{5 (x^{2}) + 5 x - 100}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 1.1.2

Factor $5$ out of $5 x$ .

$\frac{5 (x^{2}) + 5 (x) - 100}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 1.1.3

Factor $5$ out of $- 100$ .

$\frac{5 (x^{2}) + 5 x + 5 \cdot - 20}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 1.1.4

Factor $5$ out of $5 (x^{2}) + 5 x$ .

$\frac{5 (x^{2} + x) + 5 \cdot - 20}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 1.1.5

Factor $5$ out of $5 (x^{2} + x) + 5 \cdot - 20$ .

$\frac{5 (x^{2} + x - 20)}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 1.2

Factor $x^{2} + x - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{5 ((x - 4) (x + 5))}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

$\frac{5 (x - 4) (x + 5)}{x^{5} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{3}$ out of $x^{5} - 2 x^{4} - 8 x^{3}$ .

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Step 2.1.1

Factor $x^{3}$ out of $x^{5}$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} x^{2} - 2 x^{4} - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 2.1.2

Factor $x^{3}$ out of $- 2 x^{4}$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} x^{2} + x^{3} (- 2 x) - 8 x^{3}} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 2.1.3

Factor $x^{3}$ out of $- 8 x^{3}$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} x^{2} + x^{3} (- 2 x) + x^{3} \cdot - 8} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 2.1.4

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

$\frac{5 (x - 4) (x + 5)}{x^{3} (x^{2} - 2 x) + x^{3} \cdot - 8} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 2.1.5

Factor $x^{3}$ out of $x^{3} (x^{2} - 2 x) + x^{3} \cdot - 8$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} (x^{2} - 2 x - 8)} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 2.2

Factor $x^{2} - 2 x - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 2.2.2

Write the factored form using these integers.

$\frac{5 (x - 4) (x + 5)}{x^{3} ((x - 4) (x + 2))} \cdot \frac{x^{2} - 4}{15 x - 30}$

$\frac{5 (x - 4) (x + 5)}{x^{3} (x - 4) (x + 2)} \cdot \frac{x^{2} - 4}{15 x - 30}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{5 (x - 4) (x + 5)}{x^{3} (x - 4) (x + 2)} \cdot \frac{x^{2} - 2^{2}}{15 x - 30}$

Step 3.2

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 $b = 2$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} (x - 4) (x + 2)} \cdot \frac{(x + 2) (x - 2)}{15 x - 30}$

Step 4

Simplify terms.

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Step 4.1

Factor $15$ out of $15 x - 30$ .

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Step 4.1.1

Factor $15$ out of $15 x$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} (x - 4) (x + 2)} \cdot \frac{(x + 2) (x - 2)}{15 (x) - 30}$

Step 4.1.2

Factor $15$ out of $- 30$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} (x - 4) (x + 2)} \cdot \frac{(x + 2) (x - 2)}{15 x + 15 \cdot - 2}$

Step 4.1.3

Factor $15$ out of $15 x + 15 \cdot - 2$ .

$\frac{5 (x - 4) (x + 5)}{x^{3} (x - 4) (x + 2)} \cdot \frac{(x + 2) (x - 2)}{15 (x - 2)}$

Step 4.2

Combine.

$\frac{5 (x - 4) (x + 5) ((x + 2) (x - 2))}{x^{3} (x - 4) (x + 2) (15 (x - 2))}$

Step 4.3

Cancel the common factor of $5$ and $15$ .

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Step 4.3.1

Factor $5$ out of $5 (x - 4) (x + 5) ((x + 2) (x - 2))$ .

$\frac{5 (((x - 4) (x + 5)) ((x + 2) (x - 2)))}{x^{3} (x - 4) (x + 2) (15 (x - 2))}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $5$ out of $x^{3} (x - 4) (x + 2) (15 (x - 2))$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $x - 4$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Cancel the common factor of $x + 2$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 4.6

Cancel the common factor of $x - 2$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Rewrite the expression.

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

Step 4.7

Move $3$ to the left of $x^{3}$ .

$\frac{x + 5}{3 x^{3}}$