Algebra Examples

$\frac{3 x - x^{2}}{x^{2} - 16} \cdot \frac{x + 4}{x^{2} + x - 12} \div \frac{x - 10}{4 x^{3} - 64 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x - x^{2}}{x^{2} - 16} \cdot \frac{x + 4}{x^{2} + x - 12} \frac{4 x^{3} - 64 x}{x - 10}$

Step 2

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

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Step 2.1

Factor $x$ out of $3 x$ .

$\frac{x \cdot 3 - x^{2}}{x^{2} - 16} \cdot \frac{x + 4}{x^{2} + x - 12} \frac{4 x^{3} - 64 x}{x - 10}$

Step 2.2

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

$\frac{x \cdot 3 + x (- x)}{x^{2} - 16} \cdot \frac{x + 4}{x^{2} + x - 12} \frac{4 x^{3} - 64 x}{x - 10}$

Step 2.3

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

$\frac{x (3 - x)}{x^{2} - 16} \cdot \frac{x + 4}{x^{2} + x - 12} \frac{4 x^{3} - 64 x}{x - 10}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{x (3 - x)}{x^{2} - 4^{2}} \cdot \frac{x + 4}{x^{2} + x - 12} \frac{4 x^{3} - 64 x}{x - 10}$

Step 3.2

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 $b = 4$ .

$\frac{x (3 - x)}{(x + 4) (x - 4)} \cdot \frac{x + 4}{x^{2} + x - 12} \frac{4 x^{3} - 64 x}{x - 10}$

Step 4

Factor $x^{2} + x - 12$ using the AC method.

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Step 4.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 4.2

Write the factored form using these integers.

$\frac{x (3 - x)}{(x + 4) (x - 4)} \cdot \frac{x + 4}{(x - 3) (x + 4)} \frac{4 x^{3} - 64 x}{x - 10}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 4$ .

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Step 5.1.1

Cancel the common factor.

$\frac{x (3 - x)}{(x + 4) (x - 4)} \cdot \frac{x + 4}{(x - 3) (x + 4)} \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.1.2

Rewrite the expression.

$\frac{x (3 - x)}{x - 4} \cdot \frac{1}{(x - 3) (x + 4)} \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.2

Multiply $\frac{x (3 - x)}{x - 4}$ by $\frac{1}{(x - 3) (x + 4)}$ .

$\frac{x (3 - x)}{(x - 4) ((x - 3) (x + 4))} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.3

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

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Step 5.3.1

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

$\frac{x (- 1 (- 3) - x)}{(x - 4) ((x - 3) (x + 4))} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.3.2

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

$\frac{x (- 1 (- 3) - (x))}{(x - 4) ((x - 3) (x + 4))} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.3.3

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

$\frac{x (- 1 (- 3 + x))}{(x - 4) ((x - 3) (x + 4))} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.3.4

Reorder terms.

$\frac{x (- 1 (x - 3))}{(x - 4) (x - 3) (x + 4)} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.3.5

Cancel the common factor.

$\frac{x (- 1 (x - 3))}{(x - 4) (x - 3) (x + 4)} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.3.6

Rewrite the expression.

$\frac{x \cdot (- 1)}{(x - 4) (x + 4)} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.4

Simplify the expression.

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Step 5.4.1

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

$\frac{- 1 \cdot x}{(x - 4) (x + 4)} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{x}{(x - 4) (x + 4)} \cdot \frac{4 x^{3} - 64 x}{x - 10}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $4 x$ out of $4 x^{3} - 64 x$ .

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Step 6.1.1

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

$- \frac{x}{(x - 4) (x + 4)} \cdot \frac{4 x (x^{2}) - 64 x}{x - 10}$

Step 6.1.2

Factor $4 x$ out of $- 64 x$ .

$- \frac{x}{(x - 4) (x + 4)} \cdot \frac{4 x (x^{2}) + 4 x (- 16)}{x - 10}$

Step 6.1.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 16)$ .

$- \frac{x}{(x - 4) (x + 4)} \cdot \frac{4 x (x^{2} - 16)}{x - 10}$

Step 6.2

Rewrite $16$ as $4^{2}$ .

$- \frac{x}{(x - 4) (x + 4)} \cdot \frac{4 x (x^{2} - 4^{2})}{x - 10}$

Step 6.3

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 $b = 4$ .

$- \frac{x}{(x - 4) (x + 4)} \cdot \frac{4 x (x + 4) (x - 4)}{x - 10}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $(x + 4) (x - 4)$ .

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Step 7.1.1

Move the leading negative in $- \frac{x}{(x - 4) (x + 4)}$ into the numerator.

$\frac{- x}{(x - 4) (x + 4)} \cdot \frac{4 x (x + 4) (x - 4)}{x - 10}$

Step 7.1.2

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

$\frac{- x}{(x + 4) (x - 4) (1)} \cdot \frac{4 x (x + 4) (x - 4)}{x - 10}$

Step 7.1.3

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

$\frac{- x}{(x + 4) (x - 4) (1)} \cdot \frac{(x + 4) (x - 4) (4 x)}{x - 10}$

Step 7.1.4

Cancel the common factor.

$\frac{- x}{(x + 4) (x - 4) \cdot 1} \cdot \frac{(x + 4) (x - 4) (4 x)}{x - 10}$

Step 7.1.5

Rewrite the expression.

$- x \frac{4 x}{x - 10}$

Step 7.2

Combine $\frac{4 x}{x - 10}$ and $x$ .

$- \frac{4 x \cdot x}{x - 10}$

Step 8

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

$- \frac{4 (x^{1} x)}{x - 10}$

Step 9

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

$- \frac{4 (x^{1} x^{1})}{x - 10}$

Step 10

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

$- \frac{4 x^{1 + 1}}{x - 10}$

Step 11

Add $1$ and $1$ .

$- \frac{4 x^{2}}{x - 10}$