Algebra Examples

$\frac{5 x^{4} - 500 x^{2}}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $5 x^{2}$ out of $5 x^{4} - 500 x^{2}$ .

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Step 1.1.1

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

$\frac{5 x^{2} (x^{2}) - 500 x^{2}}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 1.1.2

Factor $5 x^{2}$ out of $- 500 x^{2}$ .

$\frac{5 x^{2} (x^{2}) + 5 x^{2} (- 100)}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 1.1.3

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

$\frac{5 x^{2} (x^{2} - 100)}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 1.2

Rewrite $100$ as $10^{2}$ .

$\frac{5 x^{2} (x^{2} - 10^{2})}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

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 $b = 10$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 x + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 2

Factor $15$ out of $15 x + 75$ .

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Step 2.1

Factor $15$ out of $15 x$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x) + 75} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 2.2

Factor $15$ out of $75$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 x + 15 \cdot 5} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 2.3

Factor $15$ out of $15 x + 15 \cdot 5$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{x^{2} + 11 x + 30}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 3

Factor $x^{2} + 11 x + 30$ 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 $30$ and whose sum is $11$ .

$5, 6$

Step 3.2

Write the factored form using these integers.

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{5} + 16 x^{4} + 60 x^{3}}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $x^{3}$ out of $x^{5} + 16 x^{4} + 60 x^{3}$ .

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Step 4.1.1

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

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} x^{2} + 16 x^{4} + 60 x^{3}}$

Step 4.1.2

Factor $x^{3}$ out of $16 x^{4}$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} x^{2} + x^{3} (16 x) + 60 x^{3}}$

Step 4.1.3

Factor $x^{3}$ out of $60 x^{3}$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} x^{2} + x^{3} (16 x) + x^{3} \cdot 60}$

Step 4.1.4

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (16 x)$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} (x^{2} + 16 x) + x^{3} \cdot 60}$

Step 4.1.5

Factor $x^{3}$ out of $x^{3} (x^{2} + 16 x) + x^{3} \cdot 60$ .

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} (x^{2} + 16 x + 60)}$

Step 4.2

Factor $x^{2} + 16 x + 60$ using the AC method.

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Step 4.2.1

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 $60$ and whose sum is $16$ .

$6, 10$

Step 4.2.2

Write the factored form using these integers.

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} ((x + 6) (x + 10))}$

$\frac{5 x^{2} (x + 10) (x - 10)}{15 (x + 5)} \cdot \frac{(x + 5) (x + 6)}{x^{3} (x + 6) (x + 10)}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{5 x^{2} (x + 10) (x - 10) ((x + 5) (x + 6))}{15 (x + 5) (x^{3} (x + 6) (x + 10))}$

Step 5.2

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

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Step 5.2.1

Factor $5$ out of $5 x^{2} (x + 10) (x - 10) ((x + 5) (x + 6))$ .

$\frac{5 (((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6)))}{15 (x + 5) (x^{3} (x + 6) (x + 10))}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $5$ out of $15 (x + 5) (x^{3} (x + 6) (x + 10))$ .

$\frac{5 (((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6)))}{5 (3 (x + 5) (x^{3} (x + 6) (x + 10)))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{5 (((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6)))}{5 (3 (x + 5) (x^{3} (x + 6) (x + 10)))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6))}{3 (x + 5) (x^{3} (x + 6) (x + 10))}$

Step 5.3

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.3.1

Factor $x^{2}$ out of $((x^{2} (x + 10)) (x - 10)) ((x + 5) (x + 6))$ .

$\frac{x^{2} (((x + 10) (x - 10)) ((x + 5) (x + 6)))}{3 (x + 5) (x^{3} (x + 6) (x + 10))}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $x^{2}$ out of $3 (x + 5) (x^{3} (x + 6) (x + 10))$ .

$\frac{x^{2} (((x + 10) (x - 10)) ((x + 5) (x + 6)))}{x^{2} (3 (x + 5) ((x (x + 6)) (x + 10)))}$

Step 5.3.2.2

Cancel the common factor.

$\frac{x^{2} (((x + 10) (x - 10)) ((x + 5) (x + 6)))}{x^{2} (3 (x + 5) ((x (x + 6)) (x + 10)))}$

Step 5.3.2.3

Rewrite the expression.

$\frac{((x + 10) (x - 10)) ((x + 5) (x + 6))}{3 (x + 5) ((x (x + 6)) (x + 10))}$

Step 5.4

Cancel the common factor of $x + 10$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(x + 10) (x - 10) ((x + 5) (x + 6))}{3 (x + 5) (x (x + 6) (x + 10))}$

Step 5.4.2

Rewrite the expression.

$\frac{(x - 10) ((x + 5) (x + 6))}{3 (x + 5) (x (x + 6))}$

Step 5.5

Cancel the common factor of $x + 5$ .

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Step 5.5.1

Cancel the common factor.

$\frac{(x - 10) ((x + 5) (x + 6))}{3 (x + 5) (x (x + 6))}$

Step 5.5.2

Rewrite the expression.

$\frac{(x - 10) (x + 6)}{3 (x (x + 6))}$

Step 5.6

Cancel the common factor of $x + 6$ .

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Step 5.6.1

Cancel the common factor.

$\frac{(x - 10) (x + 6)}{3 (x (x + 6))}$

Step 5.6.2

Rewrite the expression.

$\frac{x - 10}{3 (x)}$

$\frac{x - 10}{3 x}$