Precalculus Examples

$\frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}$

Step 1

Multiply the numerator and denominator of the fraction by $81$ .

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Step 1.1

Multiply $\frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}$ by $\frac{81}{81}$ .

$\frac{81}{81} \cdot \frac{\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81}}{x - \frac{1}{3}}$

Step 1.2

Combine.

$\frac{81 (\frac{1}{3} x^{3} - \frac{2}{9} x^{2} + \frac{2}{27} x - \frac{1}{81})}{81 (x - \frac{1}{3})}$

Step 2

Apply the distributive property.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) + 81 (- \frac{1}{81})}{81 x + 81 (- \frac{1}{3})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $81$ .

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Step 3.1.1

Move the leading negative in $- \frac{1}{81}$ into the numerator.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) + 81 (\frac{- 1}{81})}{81 x + 81 (- \frac{1}{3})}$

Step 3.1.2

Cancel the common factor.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) + 81 (\frac{- 1}{81})}{81 x + 81 (- \frac{1}{3})}$

Step 3.1.3

Rewrite the expression.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) - 1}{81 x + 81 (- \frac{1}{3})}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) - 1}{81 x + 81 (\frac{- 1}{3})}$

Step 3.2.2

Factor $3$ out of $81$ .

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) - 1}{81 x + 3 (27) \frac{- 1}{3}}$

Step 3.2.3

Cancel the common factor.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) - 1}{81 x + 3 \cdot 27 \frac{- 1}{3}}$

Step 3.2.4

Rewrite the expression.

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) - 1}{81 x + 27 \cdot - 1}$

Step 3.3

Multiply $27$ by $- 1$ .

$\frac{81 (\frac{1}{3} x^{3}) + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27} x) - 1}{81 x - 27}$

Step 4

Simplify the numerator.

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Step 4.1

Cancel the common factor of $3$ .

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Step 4.1.1

Factor $3$ out of $81$ .

$\frac{3 (27) \frac{1}{3} x^{3} + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.1.2

Cancel the common factor.

$\frac{3 \cdot 27 \frac{1}{3} x^{3} + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.1.3

Rewrite the expression.

$\frac{27 x^{3} + 81 (- \frac{2}{9} x^{2}) + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.2

Combine $x^{2}$ and $\frac{2}{9}$ .

$\frac{27 x^{3} + 81 (- \frac{x^{2} \cdot 2}{9}) + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.3

Cancel the common factor of $9$ .

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Step 4.3.1

Move the leading negative in $- \frac{x^{2} \cdot 2}{9}$ into the numerator.

$\frac{27 x^{3} + 81 \frac{- x^{2} \cdot 2}{9} + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.3.2

Factor $9$ out of $81$ .

$\frac{27 x^{3} + 9 (9) \frac{- x^{2} \cdot 2}{9} + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.3.3

Cancel the common factor.

$\frac{27 x^{3} + 9 \cdot 9 \frac{- x^{2} \cdot 2}{9} + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.3.4

Rewrite the expression.

$\frac{27 x^{3} + 9 (- x^{2} \cdot 2) + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.4

Multiply $2$ by $- 1$ .

$\frac{27 x^{3} + 9 (- 2 x^{2}) + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.5

Multiply $- 2$ by $9$ .

$\frac{27 x^{3} - 18 x^{2} + 81 (\frac{2}{27}) x - 1}{81 x - 27}$

Step 4.6

Cancel the common factor of $27$ .

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Step 4.6.1

Factor $27$ out of $81$ .

$\frac{27 x^{3} - 18 x^{2} + 27 (3) \frac{2}{27} x - 1}{81 x - 27}$

Step 4.6.2

Cancel the common factor.

$\frac{27 x^{3} - 18 x^{2} + 27 \cdot 3 \frac{2}{27} x - 1}{81 x - 27}$

Step 4.6.3

Rewrite the expression.

$\frac{27 x^{3} - 18 x^{2} + 3 \cdot 2 x - 1}{81 x - 27}$

Step 4.7

Multiply $3$ by $2$ .

$\frac{27 x^{3} - 18 x^{2} + 6 x - 1}{81 x - 27}$

Step 4.8

Factor $27 x^{3} - 18 x^{2} + 6 x - 1$ using the rational roots test.

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Step 4.8.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1$

$q = \pm 1, \pm 27, \pm 3, \pm 9$

Step 4.8.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.\overline{037}, \pm 0.\overline{3}, \pm 0.\overline{1}$

Step 4.8.3

Substitute $0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{3}$ is a root of the polynomial.

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Step 4.8.3.1

Substitute $0.\overline{3}$ into the polynomial.

$27 \cdot {0.\overline{3}}^{3} - 18 \cdot {0.\overline{3}}^{2} + 6 \cdot 0.\overline{3} - 1$

Step 4.8.3.2

Raise $0.\overline{3}$ to the power of $3$ .

$27 \cdot 0.\overline{037} - 18 \cdot {0.\overline{3}}^{2} + 6 \cdot 0.\overline{3} - 1$

Step 4.8.3.3

Multiply $27$ by $0.\overline{037}$ .

$1 - 18 \cdot {0.\overline{3}}^{2} + 6 \cdot 0.\overline{3} - 1$

Step 4.8.3.4

Raise $0.\overline{3}$ to the power of $2$ .

$1 - 18 \cdot 0.\overline{1} + 6 \cdot 0.\overline{3} - 1$

Step 4.8.3.5

Multiply $- 18$ by $0.\overline{1}$ .

$1 - 2 + 6 \cdot 0.\overline{3} - 1$

Step 4.8.3.6

Subtract $2$ from $1$ .

$- 1 + 6 \cdot 0.\overline{3} - 1$

Step 4.8.3.7

Multiply $6$ by $0.\overline{3}$ .

$- 1 + 2 - 1$

Step 4.8.3.8

Add $- 1$ and $2$ .

$1 - 1$

Step 4.8.3.9

Subtract $1$ from $1$ .

$0$

Step 4.8.4

Since $0.\overline{3}$ is a known root, divide the polynomial by $3 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{27 x^{3} - 18 x^{2} + 6 x - 1}{3 x - 1}$

Step 4.8.5

Divide $27 x^{3} - 18 x^{2} + 6 x - 1$ by $3 x - 1$ .

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Step 4.8.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x$

$1$

$27 x^{3}$

$18 x^{2}$

$6 x$

$1$

Step 4.8.5.2

Divide the highest order term in the dividend $27 x^{3}$ by the highest order term in divisor $3 x$ .

					$9 x^{2}$
	$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$

Step 4.8.5.3

Multiply the new quotient term by the divisor.

				$9 x^{2}$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			+	$27 x^{3}$	-	$9 x^{2}$

Step 4.8.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $27 x^{3} - 9 x^{2}$

				$9 x^{2}$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$

Step 4.8.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$9 x^{2}$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$

Step 4.8.5.6

Pull the next terms from the original dividend down into the current dividend.

				$9 x^{2}$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$

Step 4.8.5.7

Divide the highest order term in the dividend $- 9 x^{2}$ by the highest order term in divisor $3 x$ .

				$9 x^{2}$	-	$3 x$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$

Step 4.8.5.8

Multiply the new quotient term by the divisor.

				$9 x^{2}$	-	$3 x$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					-	$9 x^{2}$	+	$3 x$

Step 4.8.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{2} + 3 x$

				$9 x^{2}$	-	$3 x$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$

Step 4.8.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$9 x^{2}$	-	$3 x$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$
							+	$3 x$

Step 4.8.5.11

Pull the next terms from the original dividend down into the current dividend.

				$9 x^{2}$	-	$3 x$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$
							+	$3 x$	-	$1$

Step 4.8.5.12

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $3 x$ .

				$9 x^{2}$	-	$3 x$	+	$1$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$
							+	$3 x$	-	$1$

Step 4.8.5.13

Multiply the new quotient term by the divisor.

				$9 x^{2}$	-	$3 x$	+	$1$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$
							+	$3 x$	-	$1$
							+	$3 x$	-	$1$

Step 4.8.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 1$

				$9 x^{2}$	-	$3 x$	+	$1$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$
							+	$3 x$	-	$1$
							-	$3 x$	+	$1$

Step 4.8.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$9 x^{2}$	-	$3 x$	+	$1$
$3 x$	-	$1$		$27 x^{3}$	-	$18 x^{2}$	+	$6 x$	-	$1$
			-	$27 x^{3}$	+	$9 x^{2}$
					-	$9 x^{2}$	+	$6 x$
					+	$9 x^{2}$	-	$3 x$
							+	$3 x$	-	$1$
							-	$3 x$	+	$1$
										$0$

Step 4.8.5.16

Since the remander is $0$ , the final answer is the quotient.

$9 x^{2} - 3 x + 1$

Step 4.8.6

Write $27 x^{3} - 18 x^{2} + 6 x - 1$ as a set of factors.

$\frac{(3 x - 1) (9 x^{2} - 3 x + 1)}{81 x - 27}$

Step 5

Factor $27$ out of $81 x - 27$ .

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Step 5.1

Factor $27$ out of $81 x$ .

$\frac{(3 x - 1) (9 x^{2} - 3 x + 1)}{27 (3 x) - 27}$

Step 5.2

Factor $27$ out of $- 27$ .

$\frac{(3 x - 1) (9 x^{2} - 3 x + 1)}{27 (3 x) + 27 (- 1)}$

Step 5.3

Factor $27$ out of $27 (3 x) + 27 (- 1)$ .

$\frac{(3 x - 1) (9 x^{2} - 3 x + 1)}{27 (3 x - 1)}$

Step 6

Cancel the common factor of $3 x - 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(3 x - 1) (9 x^{2} - 3 x + 1)}{27 (3 x - 1)}$

Step 6.2

Rewrite the expression.

$\frac{9 x^{2} - 3 x + 1}{27}$

Step 7

Split the fraction $\frac{9 x^{2} - 3 x + 1}{27}$ into two fractions.

$\frac{9 x^{2} - 3 x}{27} + \frac{1}{27}$

Step 8

Split the fraction $\frac{9 x^{2} - 3 x}{27}$ into two fractions.

$\frac{9 x^{2}}{27} + \frac{- 3 x}{27} + \frac{1}{27}$

Step 9

Cancel the common factor of $9$ and $27$ .

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Step 9.1

Factor $9$ out of $9 x^{2}$ .

$\frac{9 (x^{2})}{27} + \frac{- 3 x}{27} + \frac{1}{27}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $9$ out of $27$ .

$\frac{9 x^{2}}{9 \cdot 3} + \frac{- 3 x}{27} + \frac{1}{27}$

Step 9.2.2

Cancel the common factor.

$\frac{9 x^{2}}{9 \cdot 3} + \frac{- 3 x}{27} + \frac{1}{27}$

Step 9.2.3

Rewrite the expression.

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

Step 10

Cancel the common factor of $- 3$ and $27$ .

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Step 10.1

Factor $3$ out of $- 3 x$ .

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

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $3$ out of $27$ .

$\frac{x^{2}}{3} + \frac{3 (- x)}{3 (9)} + \frac{1}{27}$

Step 10.2.2

Cancel the common factor.

$\frac{x^{2}}{3} + \frac{3 (- x)}{3 \cdot 9} + \frac{1}{27}$

Step 10.2.3

Rewrite the expression.

$\frac{x^{2}}{3} + \frac{- x}{9} + \frac{1}{27}$

Step 11

Move the negative in front of the fraction.

$\frac{x^{2}}{3} - \frac{x}{9} + \frac{1}{27}$