Algebra Examples

$\frac{x^{2} - 6 x + 9}{12 - 4 x} \cdot \frac{x^{6} - 9 x^{4}}{x^{3} - 3 x^{2}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{x^{2} - 6 x + 3^{2}}{12 - 4 x} \cdot \frac{x^{6} - 9 x^{4}}{x^{3} - 3 x^{2}}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 3 + 3^{2}}{12 - 4 x} \cdot \frac{x^{6} - 9 x^{4}}{x^{3} - 3 x^{2}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{{(x - 3)}^{2}}{12 - 4 x} \cdot \frac{x^{6} - 9 x^{4}}{x^{3} - 3 x^{2}}$

Step 2

Factor $4$ out of $12 - 4 x$ .

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Step 2.1

Factor $4$ out of $12$ .

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

Step 2.2

Factor $4$ out of $- 4 x$ .

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

Step 2.3

Factor $4$ out of $4 (3) + 4 (- x)$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Factor $x^{4}$ out of $x^{6} - 9 x^{4}$ .

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Step 3.1.1

Factor $x^{4}$ out of $x^{6}$ .

$\frac{{(x - 3)}^{2}}{4 (3 - x)} \cdot \frac{x^{4} x^{2} - 9 x^{4}}{x^{3} - 3 x^{2}}$

Step 3.1.2

Factor $x^{4}$ out of $- 9 x^{4}$ .

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

Step 3.1.3

Factor $x^{4}$ out of $x^{4} x^{2} + x^{4} \cdot - 9$ .

$\frac{{(x - 3)}^{2}}{4 (3 - x)} \cdot \frac{x^{4} (x^{2} - 9)}{x^{3} - 3 x^{2}}$

Step 3.2

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

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

Step 3.3

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 $b = 3$ .

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

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Combine.

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

Step 5

Multiply ${(x - 3)}^{2}$ by $x - 3$ by adding the exponents.

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Step 5.1

Move $x - 3$ .

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

Step 5.2

Multiply $x - 3$ by ${(x - 3)}^{2}$ .

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Step 5.2.1

Raise $x - 3$ to the power of $1$ .

$\frac{{(x - 3)}^{1} {(x - 3)}^{2} (x^{4} (x + 3))}{4 (3 - x) (x^{2} (x - 3))}$

Step 5.2.2

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

$\frac{{(x - 3)}^{1 + 2} (x^{4} (x + 3))}{4 (3 - x) (x^{2} (x - 3))}$

Step 5.3

Add $1$ and $2$ .

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

Step 6

Cancel the common factor of ${(x - 3)}^{3}$ and $x - 3$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

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

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Step 8.1

Factor $- 1$ out of $x$ .

$\frac{{(- 1 (- x) - 3)}^{2} (x^{2} (x + 3))}{4 (3 - x)}$

Step 8.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{{(- 1 (- x) - 1 (3))}^{2} (x^{2} (x + 3))}{4 (3 - x)}$

Step 8.3

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

$\frac{{(- 1 (- x + 3))}^{2} (x^{2} (x + 3))}{4 (3 - x)}$

Step 8.4

Apply the product rule to $- 1 (- x + 3)$ .

$\frac{{(- 1)}^{2} {(- x + 3)}^{2} (x^{2} (x + 3))}{4 (3 - x)}$

Step 8.5

Raise $- 1$ to the power of $2$ .

$\frac{1 {(- x + 3)}^{2} (x^{2} (x + 3))}{4 (3 - x)}$

Step 8.6

Multiply ${(- x + 3)}^{2}$ by $1$ .

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

Step 8.7

Reorder terms.

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

Step 8.8

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

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

Step 8.9

Cancel the common factors.

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Step 8.9.1

Factor $- x + 3$ out of $4 (- x + 3)$ .

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

Step 8.9.2

Cancel the common factor.

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

Step 8.9.3

Rewrite the expression.

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

Step 9

Factor $- 1$ out of $- x$ .

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

Step 10

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(- (x) - 1 (- 3)) x^{2} (x + 3)}{4}$

Step 11

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 12

Simplify the expression.

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Step 12.1

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{- 1 (x - 3) x^{2} (x + 3)}{4}$

Step 12.2

Move the negative in front of the fraction.

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

Step 12.3

Reorder factors in $- \frac{(x - 3) x^{2} (x + 3)}{4}$ .

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