Algebra Examples

$\frac{x^{5} - 12 x^{4} + 27 x^{3}}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x^{3}$ out of $x^{5} - 12 x^{4} + 27 x^{3}$ .

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Step 1.1.1

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

$\frac{x^{3} x^{2} - 12 x^{4} + 27 x^{3}}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 1.1.2

Factor $x^{3}$ out of $- 12 x^{4}$ .

$\frac{x^{3} x^{2} + x^{3} (- 12 x) + 27 x^{3}}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 1.1.3

Factor $x^{3}$ out of $27 x^{3}$ .

$\frac{x^{3} x^{2} + x^{3} (- 12 x) + x^{3} \cdot 27}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 1.1.4

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (- 12 x)$ .

$\frac{x^{3} (x^{2} - 12 x) + x^{3} \cdot 27}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 1.1.5

Factor $x^{3}$ out of $x^{3} (x^{2} - 12 x) + x^{3} \cdot 27$ .

$\frac{x^{3} (x^{2} - 12 x + 27)}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 1.2

Factor $x^{2} - 12 x + 27$ 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 $27$ and whose sum is $- 12$ .

$- 9, - 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{x^{3} ((x - 9) (x - 3))}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 x^{2} - 72}{6 x^{2} - 90 x + 324}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $2 x^{2} - 72$ .

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Step 2.1.1

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

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x^{2}) - 72}{6 x^{2} - 90 x + 324}$

Step 2.1.2

Factor $2$ out of $- 72$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 x^{2} + 2 \cdot - 36}{6 x^{2} - 90 x + 324}$

Step 2.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 36$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x^{2} - 36)}{6 x^{2} - 90 x + 324}$

Step 2.2

Rewrite $36$ as $6^{2}$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x^{2} - 6^{2})}{6 x^{2} - 90 x + 324}$

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 $a = x$ and $b = 6$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x + 6) (x - 6)}{6 x^{2} - 90 x + 324}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $6$ out of $6 x^{2} - 90 x + 324$ .

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Step 3.1.1

Factor $6$ out of $6 x^{2}$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x + 6) (x - 6)}{6 (x^{2}) - 90 x + 324}$

Step 3.1.2

Factor $6$ out of $- 90 x$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x + 6) (x - 6)}{6 (x^{2}) + 6 (- 15 x) + 324}$

Step 3.1.3

Factor $6$ out of $324$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x + 6) (x - 6)}{6 x^{2} + 6 (- 15 x) + 6 \cdot 54}$

Step 3.1.4

Factor $6$ out of $6 x^{2} + 6 (- 15 x)$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x + 6) (x - 6)}{6 (x^{2} - 15 x) + 6 \cdot 54}$

Step 3.1.5

Factor $6$ out of $6 (x^{2} - 15 x) + 6 \cdot 54$ .

$\frac{x^{3} (x - 9) (x - 3)}{x + 6} \cdot \frac{2 (x + 6) (x - 6)}{6 (x^{2} - 15 x + 54)}$

Step 3.2

Factor $x^{2} - 15 x + 54$ 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 $54$ and whose sum is $- 15$ .

$- 9, - 6$

Step 3.2.2

Write the factored form using these integers.

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

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 9$ .

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Step 4.1.1

Factor $x - 9$ out of $x^{3} (x - 9) (x - 3)$ .

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

Step 4.1.2

Factor $x - 9$ out of $6 (x - 9) (x - 6)$ .

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x + 6$ .

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Step 4.2.1

Factor $x + 6$ out of $2 (x + 6) (x - 6)$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

Combine $\frac{2 (x - 6)}{6 (x - 6)}$ and $x^{3}$ .

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

Step 4.4

Cancel the common factor of $2$ and $6$ .

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $6 (x - 6)$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 4.5

Cancel the common factor of $x - 6$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 4.6

Apply the distributive property.

$\frac{x^{3}}{3} x + \frac{x^{3}}{3} \cdot - 3$

Step 5

Multiply $\frac{x^{3}}{3} x$ .

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Step 5.1

Combine $\frac{x^{3}}{3}$ and $x$ .

$\frac{x^{3} x}{3} + \frac{x^{3}}{3} \cdot - 3$

Step 5.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 5.2.1

Multiply $x^{3}$ by $x$ .

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Step 5.2.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{3} x^{1}}{3} + \frac{x^{3}}{3} \cdot - 3$

Step 5.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3 + 1}}{3} + \frac{x^{3}}{3} \cdot - 3$

Step 5.2.2

Add $3$ and $1$ .

$\frac{x^{4}}{3} + \frac{x^{3}}{3} \cdot - 3$

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Factor $3$ out of $- 3$ .

$\frac{x^{4}}{3} + \frac{x^{3}}{3} \cdot (3 (- 1))$

Step 6.2

Cancel the common factor.

$\frac{x^{4}}{3} + \frac{x^{3}}{3} \cdot (3 \cdot - 1)$

Step 6.3

Rewrite the expression.

$\frac{x^{4}}{3} + x^{3} \cdot - 1$

Step 7

Simplify each term.

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Step 7.1

Move $- 1$ to the left of $x^{3}$ .

$\frac{x^{4}}{3} - 1 \cdot x^{3}$

Step 7.2

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$\frac{x^{4}}{3} - x^{3}$

Step 8

To write $- x^{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x^{4}}{3} - x^{3} \cdot \frac{3}{3}$

Step 9

Simplify terms.

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Step 9.1

Combine $- x^{3}$ and $\frac{3}{3}$ .

$\frac{x^{4}}{3} + \frac{- x^{3} \cdot 3}{3}$

Step 9.2

Combine the numerators over the common denominator.

$\frac{x^{4} - x^{3} \cdot 3}{3}$

Step 10

Simplify the numerator.

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Step 10.1

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

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Step 10.1.1

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

$\frac{x^{3} x - x^{3} \cdot 3}{3}$

Step 10.1.2

Factor $x^{3}$ out of $- x^{3} \cdot 3$ .

$\frac{x^{3} x + x^{3} (- 1 \cdot 3)}{3}$

Step 10.1.3

Factor $x^{3}$ out of $x^{3} x + x^{3} (- 1 \cdot 3)$ .

$\frac{x^{3} (x - 1 \cdot 3)}{3}$

Step 10.2

Multiply $- 1$ by $3$ .

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