Pre-Algebra Examples

$\frac{18 x^{7} - 9 x^{6} + 20 x^{5} - 10 x^{4}}{- 2 x^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x^{4}$ out of $18 x^{7} - 9 x^{6} + 20 x^{5} - 10 x^{4}$ .

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Step 1.1.1

Factor $x^{4}$ out of $18 x^{7}$ .

$\frac{x^{4} (18 x^{3}) - 9 x^{6} + 20 x^{5} - 10 x^{4}}{- 2 x^{4}}$

Step 1.1.2

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

$\frac{x^{4} (18 x^{3}) + x^{4} (- 9 x^{2}) + 20 x^{5} - 10 x^{4}}{- 2 x^{4}}$

Step 1.1.3

Factor $x^{4}$ out of $20 x^{5}$ .

$\frac{x^{4} (18 x^{3}) + x^{4} (- 9 x^{2}) + x^{4} (20 x) - 10 x^{4}}{- 2 x^{4}}$

Step 1.1.4

Factor $x^{4}$ out of $- 10 x^{4}$ .

$\frac{x^{4} (18 x^{3}) + x^{4} (- 9 x^{2}) + x^{4} (20 x) + x^{4} \cdot - 10}{- 2 x^{4}}$

Step 1.1.5

Factor $x^{4}$ out of $x^{4} (18 x^{3}) + x^{4} (- 9 x^{2})$ .

$\frac{x^{4} (18 x^{3} - 9 x^{2}) + x^{4} (20 x) + x^{4} \cdot - 10}{- 2 x^{4}}$

Step 1.1.6

Factor $x^{4}$ out of $x^{4} (18 x^{3} - 9 x^{2}) + x^{4} (20 x)$ .

$\frac{x^{4} (18 x^{3} - 9 x^{2} + 20 x) + x^{4} \cdot - 10}{- 2 x^{4}}$

Step 1.1.7

Factor $x^{4}$ out of $x^{4} (18 x^{3} - 9 x^{2} + 20 x) + x^{4} \cdot - 10$ .

$\frac{x^{4} (18 x^{3} - 9 x^{2} + 20 x - 10)}{- 2 x^{4}}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{x^{4} ((18 x^{3} - 9 x^{2}) + 20 x - 10)}{- 2 x^{4}}$

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x^{4} (9 x^{2} (2 x - 1) + 10 (2 x - 1))}{- 2 x^{4}}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{x^{4} (2 x - 1) (9 x^{2} + 10)}{- 2 x^{4}}$

Step 2

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

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Step 2.1

Cancel the common factor.

$\frac{x^{4} (2 x - 1) (9 x^{2} + 10)}{- 2 x^{4}}$

Step 2.2

Rewrite the expression.

$\frac{(2 x - 1) (9 x^{2} + 10)}{- 2}$

Step 3

Move the negative in front of the fraction.

$- \frac{(2 x - 1) (9 x^{2} + 10)}{2}$

Step 4

Expand $(2 x - 1) (9 x^{2} + 10)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$- \frac{2 x (9 x^{2} + 10) - 1 (9 x^{2} + 10)}{2}$

Step 4.2

Apply the distributive property.

$- \frac{2 x (9 x^{2}) + 2 x \cdot 10 - 1 (9 x^{2} + 10)}{2}$

Step 4.3

Apply the distributive property.

$- \frac{2 x (9 x^{2}) + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5

Simplify each term.

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Step 5.1

Rewrite using the commutative property of multiplication.

$- \frac{2 \cdot 9 x \cdot x^{2} + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.2

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

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Step 5.2.1

Move $x^{2}$ .

$- \frac{2 \cdot 9 (x^{2} x) + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.2.2

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

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Step 5.2.2.1

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

$- \frac{2 \cdot 9 (x^{2} x^{1}) + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.2.2.2

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

$- \frac{2 \cdot 9 x^{2 + 1} + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.2.3

Add $2$ and $1$ .

$- \frac{2 \cdot 9 x^{3} + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.3

Multiply $2$ by $9$ .

$- \frac{18 x^{3} + 2 x \cdot 10 - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.4

Multiply $10$ by $2$ .

$- \frac{18 x^{3} + 20 x - 1 (9 x^{2}) - 1 \cdot 10}{2}$

Step 5.5

Multiply $9$ by $- 1$ .

$- \frac{18 x^{3} + 20 x - 9 x^{2} - 1 \cdot 10}{2}$

Step 5.6

Multiply $- 1$ by $10$ .

$- \frac{18 x^{3} + 20 x - 9 x^{2} - 10}{2}$

Step 6

Split the fraction $\frac{18 x^{3} + 20 x - 9 x^{2} - 10}{2}$ into two fractions.

$- (\frac{18 x^{3} + 20 x - 9 x^{2}}{2} + \frac{- 10}{2})$

Step 7

Split the fraction $\frac{18 x^{3} + 20 x - 9 x^{2}}{2}$ into two fractions.

$- (\frac{18 x^{3} + 20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 8

Split the fraction $\frac{18 x^{3} + 20 x}{2}$ into two fractions.

$- (\frac{18 x^{3}}{2} + \frac{20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 9

Cancel the common factor of $18$ and $2$ .

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Step 9.1

Factor $2$ out of $18 x^{3}$ .

$- (\frac{2 (9 x^{3})}{2} + \frac{20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2$ out of $2$ .

$- (\frac{2 (9 x^{3})}{2 (1)} + \frac{20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 9.2.2

Cancel the common factor.

$- (\frac{2 (9 x^{3})}{2 \cdot 1} + \frac{20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 9.2.3

Rewrite the expression.

$- (\frac{9 x^{3}}{1} + \frac{20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 9.2.4

Divide $9 x^{3}$ by $1$ .

$- (9 x^{3} + \frac{20 x}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 10

Cancel the common factor of $20$ and $2$ .

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Step 10.1

Factor $2$ out of $20 x$ .

$- (9 x^{3} + \frac{2 (10 x)}{2} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $2$ out of $2$ .

$- (9 x^{3} + \frac{2 (10 x)}{2 (1)} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 10.2.2

Cancel the common factor.

$- (9 x^{3} + \frac{2 (10 x)}{2 \cdot 1} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 10.2.3

Rewrite the expression.

$- (9 x^{3} + \frac{10 x}{1} + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 10.2.4

Divide $10 x$ by $1$ .

$- (9 x^{3} + 10 x + \frac{- 9 x^{2}}{2} + \frac{- 10}{2})$

Step 11

Move the negative in front of the fraction.

$- (9 x^{3} + 10 x - \frac{9 x^{2}}{2} + \frac{- 10}{2})$

Step 12

Divide $- 10$ by $2$ .

$- (9 x^{3} + 10 x - \frac{9 x^{2}}{2} - 5)$