Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $x$ out of $9 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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

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

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 $a = 3$ and $b = x$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{2}$ out of $x^{4} - x^{3} - 6 x^{2}$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2.2

Write the factored form using these integers.

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Reorder terms.

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

Step 3.2.5

Cancel the common factor.

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

Step 3.2.6

Rewrite the expression.

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

Step 3.3

Simplify the expression.

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Step 3.3.1

Move $- 1$ to the left of $3 + x$ .

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

Step 3.3.2

Move the negative in front of the fraction.

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