Algebra Examples

$\frac{- 8 x^{8} + 18 x^{6}}{- 2 x^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x^{6}$ out of $- 8 x^{8} + 18 x^{6}$ .

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Step 1.1.1

Factor $2 x^{6}$ out of $- 8 x^{8}$ .

$\frac{2 x^{6} (- 4 x^{2}) + 18 x^{6}}{- 2 x^{4}}$

Step 1.1.2

Factor $2 x^{6}$ out of $18 x^{6}$ .

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

Step 1.1.3

Factor $2 x^{6}$ out of $2 x^{6} (- 4 x^{2}) + 2 x^{6} (9)$ .

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

Step 1.2

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

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

Step 1.3

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 1.4

Reorder $- {(2 x)}^{2}$ and $3^{2}$ .

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

Step 1.5

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 $b = 2 x$ .

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

Step 1.6

Multiply $2$ by $- 1$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $x^{6}$ and $x^{4}$ .

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Step 3.1

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

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

Step 3.2

Move the negative one from the denominator of $\frac{(x^{2} (3 + 2 x)) (3 - 2 x)}{- 1}$ .

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

Step 4

Rewrite $- 1 \cdot ((x^{2} (3 + 2 x)) (3 - 2 x))$ as $- ((x^{2} (3 + 2 x)) (3 - 2 x))$ .

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

Step 5

Apply the distributive property.

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

Step 6

Move $3$ to the left of $x^{2}$ .

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.1

Move $x$ .

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

Step 8.2

Multiply $x$ by $x^{2}$ .

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

Add $1$ and $2$ .

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

Step 9

Expand $(3 x^{2} + 2 x^{3}) (3 - 2 x)$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 10

Simplify and combine like terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply $3$ by $3$ .

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

Step 10.1.2

Rewrite using the commutative property of multiplication.

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

Step 10.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 10.1.3.1

Move $x$ .

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

Step 10.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 10.1.3.2.1

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

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

Step 10.1.3.2.2

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

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

Step 10.1.3.3

Add $1$ and $2$ .

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

Step 10.1.4

Multiply $3$ by $- 2$ .

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

Step 10.1.5

Multiply $3$ by $2$ .

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

Step 10.1.6

Rewrite using the commutative property of multiplication.

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

Step 10.1.7

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

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Step 10.1.7.1

Move $x$ .

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

Step 10.1.7.2

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

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Step 10.1.7.2.1

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

$- (9 x^{2} - 6 x^{3} + 6 x^{3} + 2 \cdot - 2 (x^{1} x^{3}))$

Step 10.1.7.2.2

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

$- (9 x^{2} - 6 x^{3} + 6 x^{3} + 2 \cdot - 2 x^{1 + 3})$

Step 10.1.7.3

Add $1$ and $3$ .

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

Step 10.1.8

Multiply $2$ by $- 2$ .

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

Step 10.2

Add $- 6 x^{3}$ and $6 x^{3}$ .

$- (9 x^{2} + 0 - 4 x^{4})$

Step 10.3

Add $9 x^{2}$ and $0$ .

$- (9 x^{2} - 4 x^{4})$

Step 11

Apply the distributive property.

$- (9 x^{2}) - (- 4 x^{4})$

Step 12

Multiply $9$ by $- 1$ .

$- 9 x^{2} - (- 4 x^{4})$

Step 13

Multiply $- 4$ by $- 1$ .

$- 9 x^{2} + 4 x^{4}$