Precalculus Examples

$9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} - 3 x - 1$

Step 1

Set $9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} - 3 x - 1$ equal to $0$ .

$9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} - 3 x - 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$- 3 x - 1 + 9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} = 0$

Step 2.1.2

Factor $- 1$ out of $- 3 x - 1$ .

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Step 2.1.2.1

Factor $- 1$ out of $- 3 x$ .

$- (3 x) - 1 + 9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} = 0$

Step 2.1.2.2

Rewrite $- 1$ as $- 1 (1)$ .

$- (3 x) - 1 \cdot 1 + 9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} = 0$

Step 2.1.2.3

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

$- (3 x + 1) + 9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2} = 0$

Step 2.1.3

Factor $x^{2}$ out of $9 x^{5} - 21 x^{4} + 10 x^{3} + 6 x^{2}$ .

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Step 2.1.3.1

Factor $x^{2}$ out of $9 x^{5}$ .

$- (3 x + 1) + x^{2} (9 x^{3}) - 21 x^{4} + 10 x^{3} + 6 x^{2} = 0$

Step 2.1.3.2

Factor $x^{2}$ out of $- 21 x^{4}$ .

$- (3 x + 1) + x^{2} (9 x^{3}) + x^{2} (- 21 x^{2}) + 10 x^{3} + 6 x^{2} = 0$

Step 2.1.3.3

Factor $x^{2}$ out of $10 x^{3}$ .

$- (3 x + 1) + x^{2} (9 x^{3}) + x^{2} (- 21 x^{2}) + x^{2} (10 x) + 6 x^{2} = 0$

Step 2.1.3.4

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

$- (3 x + 1) + x^{2} (9 x^{3}) + x^{2} (- 21 x^{2}) + x^{2} (10 x) + x^{2} \cdot 6 = 0$

Step 2.1.3.5

Factor $x^{2}$ out of $x^{2} (9 x^{3}) + x^{2} (- 21 x^{2})$ .

$- (3 x + 1) + x^{2} (9 x^{3} - 21 x^{2}) + x^{2} (10 x) + x^{2} \cdot 6 = 0$

Step 2.1.3.6

Factor $x^{2}$ out of $x^{2} (9 x^{3} - 21 x^{2}) + x^{2} (10 x)$ .

$- (3 x + 1) + x^{2} (9 x^{3} - 21 x^{2} + 10 x) + x^{2} \cdot 6 = 0$

Step 2.1.3.7

Factor $x^{2}$ out of $x^{2} (9 x^{3} - 21 x^{2} + 10 x) + x^{2} \cdot 6$ .

$- (3 x + 1) + x^{2} (9 x^{3} - 21 x^{2} + 10 x + 6) = 0$

Step 2.1.4

Factor.

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Step 2.1.4.1

Factor $9 x^{3} - 21 x^{2} + 10 x + 6$ using the rational roots test.

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Step 2.1.4.1.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

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

Step 2.1.4.1.2

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

$\pm 1, \pm 0.\overline{1}, \pm 0.\overline{3}, \pm 6, \pm 0.\overline{6}, \pm 2, \pm 0.\overline{2}, \pm 3$

Step 2.1.4.1.3

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 2.1.4.1.3.1

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

$9 {(- 0.\overline{3})}^{3} - 21 {(- 0.\overline{3})}^{2} + 10 \cdot - 0.\overline{3} + 6$

Step 2.1.4.1.3.2

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

$9 \cdot - 0.\overline{037} - 21 {(- 0.\overline{3})}^{2} + 10 \cdot - 0.\overline{3} + 6$

Step 2.1.4.1.3.3

Multiply $9$ by $- 0.\overline{037}$ .

$- 0.\overline{3} - 21 {(- 0.\overline{3})}^{2} + 10 \cdot - 0.\overline{3} + 6$

Step 2.1.4.1.3.4

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

$- 0.\overline{3} - 21 \cdot 0.\overline{1} + 10 \cdot - 0.\overline{3} + 6$

Step 2.1.4.1.3.5

Multiply $- 21$ by $0.\overline{1}$ .

$- 0.\overline{3} - 2.\overline{3} + 10 \cdot - 0.\overline{3} + 6$

Step 2.1.4.1.3.6

Subtract $2.\overline{3}$ from $- 0.\overline{3}$ .

$- 2.\overline{6} + 10 \cdot - 0.\overline{3} + 6$

Step 2.1.4.1.3.7

Multiply $10$ by $- 0.\overline{3}$ .

$- 2.\overline{6} - 3.\overline{3} + 6$

Step 2.1.4.1.3.8

Subtract $3.\overline{3}$ from $- 2.\overline{6}$ .

$- 6 + 6$

Step 2.1.4.1.3.9

Add $- 6$ and $6$ .

$0$

Step 2.1.4.1.4

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.

$\frac{9 x^{3} - 21 x^{2} + 10 x + 6}{3 x + 1}$

Step 2.1.4.1.5

Divide $9 x^{3} - 21 x^{2} + 10 x + 6$ by $3 x + 1$ .

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Step 2.1.4.1.5.1

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

$3 x$

$1$

$9 x^{3}$

$21 x^{2}$

$10 x$

$6$

Step 2.1.4.1.5.2

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

					$3 x^{2}$
	$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$

Step 2.1.4.1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			+	$9 x^{3}$	+	$3 x^{2}$

Step 2.1.4.1.5.4

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$

Step 2.1.4.1.5.5

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$

Step 2.1.4.1.5.6

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$

Step 2.1.4.1.5.7

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$

Step 2.1.4.1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					-	$24 x^{2}$	-	$8 x$

Step 2.1.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} - 8 x$

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$

Step 2.1.4.1.5.10

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

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$
							+	$18 x$

Step 2.1.4.1.5.11

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

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$
							+	$18 x$	+	$6$

Step 2.1.4.1.5.12

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$8 x$	+	$6$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$
							+	$18 x$	+	$6$

Step 2.1.4.1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$8 x$	+	$6$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$
							+	$18 x$	+	$6$
							+	$18 x$	+	$6$

Step 2.1.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $18 x + 6$

				$3 x^{2}$	-	$8 x$	+	$6$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$
							+	$18 x$	+	$6$
							-	$18 x$	-	$6$

Step 2.1.4.1.5.15

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

				$3 x^{2}$	-	$8 x$	+	$6$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	+	$10 x$	+	$6$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	+	$10 x$
					+	$24 x^{2}$	+	$8 x$
							+	$18 x$	+	$6$
							-	$18 x$	-	$6$
										$0$

Step 2.1.4.1.5.16

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

$3 x^{2} - 8 x + 6$

Step 2.1.4.1.6

Write $9 x^{3} - 21 x^{2} + 10 x + 6$ as a set of factors.

$- (3 x + 1) + x^{2} ((3 x + 1) (3 x^{2} - 8 x + 6)) = 0$

Step 2.1.4.2

Remove unnecessary parentheses.

$- (3 x + 1) + x^{2} (3 x + 1) (3 x^{2} - 8 x + 6) = 0$

Step 2.1.5

Factor $3 x + 1$ out of $- (3 x + 1) + x^{2} (3 x + 1) (3 x^{2} - 8 x + 6)$ .

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Step 2.1.5.1

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

$(3 x + 1) \cdot - 1 + x^{2} (3 x + 1) (3 x^{2} - 8 x + 6) = 0$

Step 2.1.5.2

Factor $3 x + 1$ out of $x^{2} (3 x + 1) (3 x^{2} - 8 x + 6)$ .

$(3 x + 1) \cdot - 1 + (3 x + 1) (x^{2} (3 x^{2} - 8 x + 6)) = 0$

Step 2.1.5.3

Factor $3 x + 1$ out of $(3 x + 1) \cdot - 1 + (3 x + 1) (x^{2} (3 x^{2} - 8 x + 6))$ .

$(3 x + 1) (- 1 + x^{2} (3 x^{2} - 8 x + 6)) = 0$

Step 2.1.6

Apply the distributive property.

$(3 x + 1) (- 1 + x^{2} (3 x^{2}) + x^{2} (- 8 x) + x^{2} \cdot 6) = 0$

Step 2.1.7

Simplify.

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Step 2.1.7.1

Rewrite using the commutative property of multiplication.

$(3 x + 1) (- 1 + 3 x^{2} x^{2} + x^{2} (- 8 x) + x^{2} \cdot 6) = 0$

Step 2.1.7.2

Rewrite using the commutative property of multiplication.

$(3 x + 1) (- 1 + 3 x^{2} x^{2} - 8 x^{2} x + x^{2} \cdot 6) = 0$

Step 2.1.7.3

Move $6$ to the left of $x^{2}$ .

$(3 x + 1) (- 1 + 3 x^{2} x^{2} - 8 x^{2} x + 6 \cdot x^{2}) = 0$

Step 2.1.8

Simplify each term.

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Step 2.1.8.1

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

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Step 2.1.8.1.1

Move $x^{2}$ .

$(3 x + 1) (- 1 + 3 (x^{2} x^{2}) - 8 x^{2} x + 6 \cdot x^{2}) = 0$

Step 2.1.8.1.2

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

$(3 x + 1) (- 1 + 3 x^{2 + 2} - 8 x^{2} x + 6 \cdot x^{2}) = 0$

Step 2.1.8.1.3

Add $2$ and $2$ .

$(3 x + 1) (- 1 + 3 x^{4} - 8 x^{2} x + 6 \cdot x^{2}) = 0$

Step 2.1.8.2

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

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Step 2.1.8.2.1

Move $x$ .

$(3 x + 1) (- 1 + 3 x^{4} - 8 (x \cdot x^{2}) + 6 \cdot x^{2}) = 0$

Step 2.1.8.2.2

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

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Step 2.1.8.2.2.1

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

$(3 x + 1) (- 1 + 3 x^{4} - 8 (x \cdot x^{2}) + 6 \cdot x^{2}) = 0$

Step 2.1.8.2.2.2

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

$(3 x + 1) (- 1 + 3 x^{4} - 8 x^{1 + 2} + 6 \cdot x^{2}) = 0$

Step 2.1.8.2.3

Add $1$ and $2$ .

$(3 x + 1) (- 1 + 3 x^{4} - 8 x^{3} + 6 \cdot x^{2}) = 0$

$(3 x + 1) (- 1 + 3 x^{4} - 8 x^{3} + 6 x^{2}) = 0$

Step 2.1.9

Reorder terms.

$(3 x + 1) (3 x^{4} - 8 x^{3} + 6 x^{2} - 1) = 0$

Step 2.1.10

Factor.

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Step 2.1.10.1

Rewrite $3 x^{4} - 8 x^{3} + 6 x^{2} - 1$ in a factored form.

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Step 2.1.10.1.1

Factor $3 x^{4} - 8 x^{3} + 6 x^{2} - 1$ using the rational roots test.

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Step 2.1.10.1.1.1

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.

$p = \pm 1$

$q = \pm 1, \pm 3$

Step 2.1.10.1.1.2

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

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

Step 2.1.10.1.1.3

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 2.1.10.1.1.3.1

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

$3 {(- 0.\overline{3})}^{4} - 8 {(- 0.\overline{3})}^{3} + 6 {(- 0.\overline{3})}^{2} - 1$

Step 2.1.10.1.1.3.2

Raise $- 0.\overline{3}$ to the power of $4$ .

$3 \cdot 0.01234567 - 8 {(- 0.\overline{3})}^{3} + 6 {(- 0.\overline{3})}^{2} - 1$

Step 2.1.10.1.1.3.3

Multiply $3$ by $0.01234567$ .

$0.\overline{037} - 8 {(- 0.\overline{3})}^{3} + 6 {(- 0.\overline{3})}^{2} - 1$

Step 2.1.10.1.1.3.4

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

$0.\overline{037} - 8 \cdot - 0.\overline{037} + 6 {(- 0.\overline{3})}^{2} - 1$

Step 2.1.10.1.1.3.5

Multiply $- 8$ by $- 0.\overline{037}$ .

$0.\overline{037} + 0.\overline{296} + 6 {(- 0.\overline{3})}^{2} - 1$

Step 2.1.10.1.1.3.6

Add $0.\overline{037}$ and $0.\overline{296}$ .

$0.\overline{3} + 6 {(- 0.\overline{3})}^{2} - 1$

Step 2.1.10.1.1.3.7

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

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

Step 2.1.10.1.1.3.8

Multiply $6$ by $0.\overline{1}$ .

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

Step 2.1.10.1.1.3.9

Add $0.\overline{3}$ and $0.\overline{6}$ .

$1 - 1$

Step 2.1.10.1.1.3.10

Subtract $1$ from $1$ .

$0$

Step 2.1.10.1.1.4

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.

$\frac{3 x^{4} - 8 x^{3} + 6 x^{2} - 1}{3 x + 1}$

Step 2.1.10.1.1.5

Divide $3 x^{4} - 8 x^{3} + 6 x^{2} - 1$ by $3 x + 1$ .

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Step 2.1.10.1.1.5.1

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

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

Step 2.1.10.1.1.5.2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

Step 2.1.10.1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

Step 2.1.10.1.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + x^{3}$

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

Step 2.1.10.1.1.5.5

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

Step 2.1.10.1.1.5.6

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

$x^{3}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

Step 2.1.10.1.1.5.7

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

Step 2.1.10.1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

Step 2.1.10.1.1.5.9

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

Step 2.1.10.1.1.5.10

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

Step 2.1.10.1.1.5.11

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

Step 2.1.10.1.1.5.12

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

Step 2.1.10.1.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

Step 2.1.10.1.1.5.14

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

Step 2.1.10.1.1.5.15

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

Step 2.1.10.1.1.5.16

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

$1$

Step 2.1.10.1.1.5.17

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

$1$

Step 2.1.10.1.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

$1$

$3 x$

$1$

Step 2.1.10.1.1.5.19

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

$1$

$3 x$

$1$

Step 2.1.10.1.1.5.20

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

$x^{3}$

$3 x^{2}$

$3 x$

$1$

$3 x$

$1$

$3 x^{4}$

$8 x^{3}$

$6 x^{2}$

$0 x$

$1$

$3 x^{4}$

$x^{3}$

$9 x^{3}$

$6 x^{2}$

$9 x^{3}$

$3 x^{2}$

$9 x^{2}$

$0 x$

$9 x^{2}$

$3 x$

$1$

$3 x$

$1$

$0$

Step 2.1.10.1.1.5.21

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

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

Step 2.1.10.1.1.6

Write $3 x^{4} - 8 x^{3} + 6 x^{2} - 1$ as a set of factors.

$(3 x + 1) ((3 x + 1) (x^{3} - 3 x^{2} + 3 x - 1)) = 0$

Step 2.1.10.1.2

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

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Step 2.1.10.1.2.1

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.

$p = \pm 1$

$q = \pm 1$

Step 2.1.10.1.2.2

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

$\pm 1$

Step 2.1.10.1.2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.1.10.1.2.3.1

Substitute $1$ into the polynomial.

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

Step 2.1.10.1.2.3.2

Raise $1$ to the power of $3$ .

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

Step 2.1.10.1.2.3.3

Raise $1$ to the power of $2$ .

$1 - 3 \cdot 1 + 3 \cdot 1 - 1$

Step 2.1.10.1.2.3.4

Multiply $- 3$ by $1$ .

$1 - 3 + 3 \cdot 1 - 1$

Step 2.1.10.1.2.3.5

Subtract $3$ from $1$ .

$- 2 + 3 \cdot 1 - 1$

Step 2.1.10.1.2.3.6

Multiply $3$ by $1$ .

$- 2 + 3 - 1$

Step 2.1.10.1.2.3.7

Add $- 2$ and $3$ .

$1 - 1$

Step 2.1.10.1.2.3.8

Subtract $1$ from $1$ .

$0$

Step 2.1.10.1.2.4

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

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

Step 2.1.10.1.2.5

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

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Step 2.1.10.1.2.5.1

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$3 x$

$1$

Step 2.1.10.1.2.5.2

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

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

Step 2.1.10.1.2.5.3

Multiply the new quotient term by the divisor.

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

Step 2.1.10.1.2.5.4

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

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

Step 2.1.10.1.2.5.5

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

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

Step 2.1.10.1.2.5.6

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

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

Step 2.1.10.1.2.5.7

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

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

Step 2.1.10.1.2.5.8

Multiply the new quotient term by the divisor.

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

Step 2.1.10.1.2.5.9

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

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

Step 2.1.10.1.2.5.10

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

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

Step 2.1.10.1.2.5.11

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

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

Step 2.1.10.1.2.5.12

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

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

Step 2.1.10.1.2.5.13

Multiply the new quotient term by the divisor.

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

Step 2.1.10.1.2.5.14

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

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

Step 2.1.10.1.2.5.15

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

				$x^{2}$	-	$2 x$	+	$1$
$x$	-	$1$		$x^{3}$	-	$3 x^{2}$	+	$3 x$	-	$1$
			-	$x^{3}$	+	$x^{2}$
					-	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$2 x$
							+	$x$	-	$1$
							-	$x$	+	$1$
										$0$

Step 2.1.10.1.2.5.16

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

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

Step 2.1.10.1.2.6

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

$(3 x + 1) ((3 x + 1) ((x - 1) (x^{2} - 2 x + 1))) = 0$

Step 2.1.10.1.3

Factor using the perfect square rule.

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Step 2.1.10.1.3.1

Rewrite $1$ as $1^{2}$ .

$(3 x + 1) ((3 x + 1) ((x - 1) (x^{2} - 2 x + 1^{2}))) = 0$

Step 2.1.10.1.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 2.1.10.1.3.3

Rewrite the polynomial.

$(3 x + 1) ((3 x + 1) ((x - 1) (x^{2} - 2 \cdot x \cdot 1 + 1^{2}))) = 0$

Step 2.1.10.1.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$(3 x + 1) ((3 x + 1) ((x - 1) {(x - 1)}^{2})) = 0$

Step 2.1.10.1.4

Combine like factors.

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Step 2.1.10.1.4.1

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

$(3 x + 1) ((3 x + 1) ((x - 1) {(x - 1)}^{2})) = 0$

Step 2.1.10.1.4.2

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

$(3 x + 1) ((3 x + 1) {(x - 1)}^{1 + 2}) = 0$

Step 2.1.10.1.4.3

Add $1$ and $2$ .

$(3 x + 1) ((3 x + 1) {(x - 1)}^{3}) = 0$

Step 2.1.10.2

Remove unnecessary parentheses.

$(3 x + 1) (3 x + 1) {(x - 1)}^{3} = 0$

Step 2.1.11

Combine exponents.

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Step 2.1.11.1

Raise $3 x + 1$ to the power of $1$ .

$(3 x + 1) (3 x + 1) {(x - 1)}^{3} = 0$

Step 2.1.11.2

Raise $3 x + 1$ to the power of $1$ .

$(3 x + 1) (3 x + 1) {(x - 1)}^{3} = 0$

Step 2.1.11.3

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

${(3 x + 1)}^{1 + 1} {(x - 1)}^{3} = 0$

Step 2.1.11.4

Add $1$ and $1$ .

${(3 x + 1)}^{2} {(x - 1)}^{3} = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(3 x + 1)}^{2} = 0$

${(x - 1)}^{3} = 0$

Step 2.3

Set ${(3 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set ${(3 x + 1)}^{2}$ equal to $0$ .

${(3 x + 1)}^{2} = 0$

Step 2.3.2

Solve ${(3 x + 1)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 2.3.2.2

Solve for $x$ .

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Step 2.3.2.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 2.3.2.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 2.3.2.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.3.2.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 2.3.2.2.2.3

Simplify the right side.

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Step 2.3.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 2.4

Set ${(x - 1)}^{3}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x - 1)}^{3}$ equal to $0$ .

${(x - 1)}^{3} = 0$

Step 2.4.2

Solve ${(x - 1)}^{3} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

The final solution is all the values that make ${(3 x + 1)}^{2} {(x - 1)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - \frac{1}{3}$ (Multiplicity of $2$ )

$x = 1$ (Multiplicity of $3$ )

$x = - \frac{1}{3}$ (Multiplicity of $2$ )

$x = 1$ (Multiplicity of $3$ )

Step 3