Algebra Examples

$\frac{9 x^{4} - 72 x}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $9 x$ out of $9 x^{4} - 72 x$ .

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Step 1.1.1

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

$\frac{9 x (x^{3}) - 72 x}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1.1.2

Factor $9 x$ out of $- 72 x$ .

$\frac{9 x (x^{3}) + 9 x (- 8)}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1.1.3

Factor $9 x$ out of $9 x (x^{3}) + 9 x (- 8)$ .

$\frac{9 x (x^{3} - 8)}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1.2

Rewrite $8$ as $2^{3}$ .

$\frac{9 x (x^{3} - 2^{3})}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

$\frac{9 x ((x - 2) (x^{2} + x \cdot 2 + 2^{2}))}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1.4

Simplify.

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Step 1.4.1

Move $2$ to the left of $x$ .

$\frac{9 x ((x - 2) (x^{2} + 2 \cdot x + 2^{2}))}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 1.4.2

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

$\frac{9 x ((x - 2) (x^{2} + 2 x + 4))}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 x^{2} - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

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

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x^{2}) - 12} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 2.1.2

Factor $3$ out of $- 12$ .

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 x^{2} + 3 \cdot - 4} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 2.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 4$ .

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x^{2} - 4)} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 2.2

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

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x^{2} - 2^{2})} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

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 $b = 2$ .

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{x^{2} + x - 2}{4 x^{3} + 8 x^{2} + 16 x}$

Step 3

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

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Step 3.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 3.2

Write the factored form using these integers.

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{(x - 1) (x + 2)}{4 x^{3} + 8 x^{2} + 16 x}$

Step 4

Simplify terms.

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Step 4.1

Factor $4 x$ out of $4 x^{3} + 8 x^{2} + 16 x$ .

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Step 4.1.1

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

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{(x - 1) (x + 2)}{4 x (x^{2}) + 8 x^{2} + 16 x}$

Step 4.1.2

Factor $4 x$ out of $8 x^{2}$ .

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{(x - 1) (x + 2)}{4 x (x^{2}) + 4 x (2 x) + 16 x}$

Step 4.1.3

Factor $4 x$ out of $16 x$ .

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{(x - 1) (x + 2)}{4 x (x^{2}) + 4 x (2 x) + 4 x (4)}$

Step 4.1.4

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

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{(x - 1) (x + 2)}{4 x (x^{2} + 2 x) + 4 x (4)}$

Step 4.1.5

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

$\frac{9 x (x - 2) (x^{2} + 2 x + 4)}{3 (x + 2) (x - 2)} \cdot \frac{(x - 1) (x + 2)}{4 x (x^{2} + 2 x + 4)}$

Step 4.2

Combine.

$\frac{9 x (x - 2) (x^{2} + 2 x + 4) ((x - 1) (x + 2))}{3 (x + 2) (x - 2) (4 x (x^{2} + 2 x + 4))}$

Step 4.3

Cancel the common factor of $9$ and $3$ .

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Step 4.3.1

Factor $3$ out of $9 x (x - 2) (x^{2} + 2 x + 4) ((x - 1) (x + 2))$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $x$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Cancel the common factor of $x - 2$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 4.6

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

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Rewrite the expression.

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

Step 4.7

Cancel the common factor of $x + 2$ .

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Step 4.7.1

Cancel the common factor.

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

Step 4.7.2

Rewrite the expression.

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

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