Precalculus Examples

$\frac{6 x^{4} - 4 x^{3} - 27 x^{2} + 18 x}{x - \frac{2}{3}}$

Step 1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.1

Multiply $\frac{6 x^{4} - 4 x^{3} - 27 x^{2} + 18 x}{x - \frac{2}{3}}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{6 x^{4} - 4 x^{3} - 27 x^{2} + 18 x}{x - \frac{2}{3}}$

Step 1.2

Combine.

$\frac{3 (6 x^{4} - 4 x^{3} - 27 x^{2} + 18 x)}{3 (x - \frac{2}{3})}$

Step 2

Apply the distributive property.

$\frac{3 (6 x^{4}) + 3 (- 4 x^{3}) + 3 (- 27 x^{2}) + 3 (18 x)}{3 x + 3 (- \frac{2}{3})}$

Step 3

Cancel the common factor of $3$ .

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Step 3.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{3 (6 x^{4}) + 3 (- 4 x^{3}) + 3 (- 27 x^{2}) + 3 (18 x)}{3 x + 3 (\frac{- 2}{3})}$

Step 3.2

Cancel the common factor.

$\frac{3 (6 x^{4}) + 3 (- 4 x^{3}) + 3 (- 27 x^{2}) + 3 (18 x)}{3 x + 3 (\frac{- 2}{3})}$

Step 3.3

Rewrite the expression.

$\frac{3 (6 x^{4}) + 3 (- 4 x^{3}) + 3 (- 27 x^{2}) + 3 (18 x)}{3 x - 2}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $3 x$ out of $3 \cdot 6 x^{4} + 3 (- 4 x^{3}) + 3 (- 27 x^{2}) + 3 \cdot 18 x$ .

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Step 4.1.1

Factor $3 x$ out of $3 \cdot 6 x^{4}$ .

$\frac{3 x (6 x^{3}) + 3 (- 4 x^{3}) + 3 (- 27 x^{2}) + 3 \cdot 18 x}{3 x - 2}$

Step 4.1.2

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

$\frac{3 x (6 x^{3}) + 3 x (- 4 x^{2}) + 3 (- 27 x^{2}) + 3 \cdot 18 x}{3 x - 2}$

Step 4.1.3

Factor $3 x$ out of $3 (- 27 x^{2})$ .

$\frac{3 x (6 x^{3}) + 3 x (- 4 x^{2}) + 3 x (- 27 x) + 3 \cdot 18 x}{3 x - 2}$

Step 4.1.4

Factor $3 x$ out of $3 \cdot 18 x$ .

$\frac{3 x (6 x^{3}) + 3 x (- 4 x^{2}) + 3 x (- 27 x) + 3 x (18)}{3 x - 2}$

Step 4.1.5

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

$\frac{3 x (6 x^{3} - 4 x^{2}) + 3 x (- 27 x) + 3 x (18)}{3 x - 2}$

Step 4.1.6

Factor $3 x$ out of $3 x (6 x^{3} - 4 x^{2}) + 3 x (- 27 x)$ .

$\frac{3 x (6 x^{3} - 4 x^{2} - 27 x) + 3 x (18)}{3 x - 2}$

Step 4.1.7

Factor $3 x$ out of $3 x (6 x^{3} - 4 x^{2} - 27 x) + 3 x (18)$ .

$\frac{3 x (6 x^{3} - 4 x^{2} - 27 x + 18)}{3 x - 2}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{3 x ((6 x^{3} - 4 x^{2}) - 27 x + 18)}{3 x - 2}$

Step 4.2.2

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

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 5

Cancel the common factor of $3 x - 2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $3 x (2 x^{2} - 9)$ by $1$ .

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

Step 6

Apply the distributive property.

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

Multiply $- 9$ by $3$ .

$3 \cdot 2 x \cdot x^{2} - 27 x$

Step 9

Simplify each term.

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Step 9.1

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

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Step 9.1.1

Move $x^{2}$ .

$3 \cdot 2 (x^{2} x) - 27 x$

Step 9.1.2

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

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Step 9.1.2.1

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

$3 \cdot 2 (x^{2} x^{1}) - 27 x$

Step 9.1.2.2

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

$3 \cdot 2 x^{2 + 1} - 27 x$

Step 9.1.3

Add $2$ and $1$ .

$3 \cdot 2 x^{3} - 27 x$

Step 9.2

Multiply $3$ by $2$ .

$6 x^{3} - 27 x$