Algebra Examples

$\frac{15 {(x + 6)}^{2}}{15 x^{2} - 60 x - 900} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 1

Simplify the denominator.

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Step 1.1

Factor $15$ out of $15 x^{2} - 60 x - 900$ .

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Step 1.1.1

Factor $15$ out of $15 x^{2}$ .

$\frac{15 {(x + 6)}^{2}}{15 (x^{2}) - 60 x - 900} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 1.1.2

Factor $15$ out of $- 60 x$ .

$\frac{15 {(x + 6)}^{2}}{15 (x^{2}) + 15 (- 4 x) - 900} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 1.1.3

Factor $15$ out of $- 900$ .

$\frac{15 {(x + 6)}^{2}}{15 x^{2} + 15 (- 4 x) + 15 \cdot - 60} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 1.1.4

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

$\frac{15 {(x + 6)}^{2}}{15 (x^{2} - 4 x) + 15 \cdot - 60} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 1.1.5

Factor $15$ out of $15 (x^{2} - 4 x) + 15 \cdot - 60$ .

$\frac{15 {(x + 6)}^{2}}{15 (x^{2} - 4 x - 60)} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 1.2

Factor $x^{2} - 4 x - 60$ 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 $- 60$ and whose sum is $- 4$ .

$- 10, 6$

Step 1.2.2

Write the factored form using these integers.

$\frac{15 {(x + 6)}^{2}}{15 ((x - 10) (x + 6))} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{5} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $x^{3}$ out of $x^{5} - 7 x^{4} - 30 x^{3}$ .

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Step 2.1.1

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

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} x^{2} - 7 x^{4} - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 2.1.2

Factor $x^{3}$ out of $- 7 x^{4}$ .

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} x^{2} + x^{3} (- 7 x) - 30 x^{3}}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 2.1.3

Factor $x^{3}$ out of $- 30 x^{3}$ .

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} x^{2} + x^{3} (- 7 x) + x^{3} \cdot - 30}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 2.1.4

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (- 7 x)$ .

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x^{2} - 7 x) + x^{3} \cdot - 30}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 2.1.5

Factor $x^{3}$ out of $x^{3} (x^{2} - 7 x) + x^{3} \cdot - 30$ .

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x^{2} - 7 x - 30)}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 2.2

Factor $x^{2} - 7 x - 30$ 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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} ((x - 10) (x + 3))}{x^{4} + 9 x^{3} + 18 x^{2}}$

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x - 10) (x + 3)}{x^{4} + 9 x^{3} + 18 x^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $x^{2}$ out of $x^{4} + 9 x^{3} + 18 x^{2}$ .

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Step 3.1.1

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

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x - 10) (x + 3)}{x^{2} x^{2} + 9 x^{3} + 18 x^{2}}$

Step 3.1.2

Factor $x^{2}$ out of $9 x^{3}$ .

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x - 10) (x + 3)}{x^{2} x^{2} + x^{2} (9 x) + 18 x^{2}}$

Step 3.1.3

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

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x - 10) (x + 3)}{x^{2} x^{2} + x^{2} (9 x) + x^{2} \cdot 18}$

Step 3.1.4

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

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x - 10) (x + 3)}{x^{2} (x^{2} + 9 x) + x^{2} \cdot 18}$

Step 3.1.5

Factor $x^{2}$ out of $x^{2} (x^{2} + 9 x) + x^{2} \cdot 18$ .

$\frac{15 {(x + 6)}^{2}}{15 (x - 10) (x + 6)} \cdot \frac{x^{3} (x - 10) (x + 3)}{x^{2} (x^{2} + 9 x + 18)}$

Step 3.2

Factor $x^{2} + 9 x + 18$ using the AC method.

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Step 3.2.1

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 $18$ and whose sum is $9$ .

$3, 6$

Step 3.2.2

Write the factored form using these integers.

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

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $15$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Cancel the common factor of ${(x + 6)}^{2}$ and $x + 6$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $x + 6$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

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

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

Step 4.5.2.2

Cancel the common factor.

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

Step 4.5.2.3

Rewrite the expression.

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

Step 4.6

Cancel the common factor of $x - 10$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Rewrite the expression.

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

Step 4.7

Cancel the common factor of $x + 3$ .

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Step 4.7.1

Cancel the common factor.

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

Step 4.7.2

Divide $x$ by $1$ .

$x$