Precalculus Examples

$x^{3} + 9 x^{2} + 14 x - 24 = (x + 6)$

Step 1

Subtract $(x + 6)$ from both sides of the equation.

$x^{3} + 9 x^{2} + 14 x - 24 - (x + 6) = 0$

Step 2

Simplify $x^{3} + 9 x^{2} + 14 x - 24 - (x + 6)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$x^{3} + 9 x^{2} + 14 x - 24 - x - 1 \cdot 6 = 0$

Step 2.1.2

Multiply $- 1$ by $6$ .

$x^{3} + 9 x^{2} + 14 x - 24 - x - 6 = 0$

Step 2.2

Subtract $x$ from $14 x$ .

$x^{3} + 9 x^{2} + 13 x - 24 - 6 = 0$

Step 2.3

Subtract $6$ from $- 24$ .

$x^{3} + 9 x^{2} + 13 x - 30 = 0$

Step 3

Factor $x^{3} + 9 x^{2} + 13 x - 30$ using the rational roots test.

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Step 3.1

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.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

$q = \pm 1$

Step 3.2

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

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

Step 3.3

Substitute $- 6$ and simplify the expression. In this case, the expression is equal to $0$ so $- 6$ is a root of the polynomial.

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Step 3.3.1

Substitute $- 6$ into the polynomial.

${(- 6)}^{3} + 9 {(- 6)}^{2} + 13 \cdot - 6 - 30$

Step 3.3.2

Raise $- 6$ to the power of $3$ .

$- 216 + 9 {(- 6)}^{2} + 13 \cdot - 6 - 30$

Step 3.3.3

Raise $- 6$ to the power of $2$ .

$- 216 + 9 \cdot 36 + 13 \cdot - 6 - 30$

Step 3.3.4

Multiply $9$ by $36$ .

$- 216 + 324 + 13 \cdot - 6 - 30$

Step 3.3.5

Add $- 216$ and $324$ .

$108 + 13 \cdot - 6 - 30$

Step 3.3.6

Multiply $13$ by $- 6$ .

$108 - 78 - 30$

Step 3.3.7

Subtract $78$ from $108$ .

$30 - 30$

Step 3.3.8

Subtract $30$ from $30$ .

$0$

Step 3.4

Since $- 6$ is a known root, divide the polynomial by $x + 6$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 9 x^{2} + 13 x - 30}{x + 6}$

Step 3.5

Divide $x^{3} + 9 x^{2} + 13 x - 30$ by $x + 6$ .

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Step 3.5.1

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

$x$

$6$

$x^{3}$

$9 x^{2}$

$13 x$

$30$

Step 3.5.2

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

					$x^{2}$
	$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$

Step 3.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			+	$x^{3}$	+	$6 x^{2}$

Step 3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 6 x^{2}$

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$

Step 3.5.5

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$

Step 3.5.6

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$

Step 3.5.7

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$

Step 3.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					+	$3 x^{2}$	+	$18 x$

Step 3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} + 18 x$

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$

Step 3.5.10

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$
							-	$5 x$

Step 3.5.11

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$
							-	$5 x$	-	$30$

Step 3.5.12

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$5$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$
							-	$5 x$	-	$30$

Step 3.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$5$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$
							-	$5 x$	-	$30$
							-	$5 x$	-	$30$

Step 3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x - 30$

				$x^{2}$	+	$3 x$	-	$5$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$
							-	$5 x$	-	$30$
							+	$5 x$	+	$30$

Step 3.5.15

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

				$x^{2}$	+	$3 x$	-	$5$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$13 x$	-	$30$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$18 x$
							-	$5 x$	-	$30$
							+	$5 x$	+	$30$
										$0$

Step 3.5.16

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

$x^{2} + 3 x - 5$

Step 3.6

Write $x^{3} + 9 x^{2} + 13 x - 30$ as a set of factors.

$(x + 6) (x^{2} + 3 x - 5) = 0$

Step 4

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

$x + 6 = 0$

$x^{2} + 3 x - 5 = 0$

Step 5

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 5.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 6

Set $x^{2} + 3 x - 5$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} + 3 x - 5$ equal to $0$ .

$x^{2} + 3 x - 5 = 0$

Step 6.2

Solve $x^{2} + 3 x - 5 = 0$ for $x$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.2.3.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{- 3 \pm \sqrt{9 + 20}}{2 \cdot 1}$

Step 6.2.3.1.3

Add $9$ and $20$ .

$x = \frac{- 3 \pm \sqrt{29}}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{29}}{2}$

Step 6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.2.4.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{- 3 \pm \sqrt{9 + 20}}{2 \cdot 1}$

Step 6.2.4.1.3

Add $9$ and $20$ .

$x = \frac{- 3 \pm \sqrt{29}}{2 \cdot 1}$

Step 6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{29}}{2}$

Step 6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{29}}{2}$

Step 6.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{29}}{2}$

Step 6.2.4.5

Factor $- 1$ out of $\sqrt{29}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{29})}{2}$

Step 6.2.4.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{29})$ .

$x = \frac{- 1 (3 - \sqrt{29})}{2}$

Step 6.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{29}}{2}$

Step 6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.2.5.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{- 3 \pm \sqrt{9 + 20}}{2 \cdot 1}$

Step 6.2.5.1.3

Add $9$ and $20$ .

$x = \frac{- 3 \pm \sqrt{29}}{2 \cdot 1}$

Step 6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{29}}{2}$

Step 6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{29}}{2}$

Step 6.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{29}}{2}$

Step 6.2.5.5

Factor $- 1$ out of $- \sqrt{29}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{29})}{2}$

Step 6.2.5.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{29})$ .

$x = \frac{- 1 (3 + \sqrt{29})}{2}$

Step 6.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{29}}{2}$

Step 6.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{29}}{2}, - \frac{3 + \sqrt{29}}{2}$

Step 7

The final solution is all the values that make $(x + 6) (x^{2} + 3 x - 5) = 0$ true.

$x = - 6, - \frac{3 - \sqrt{29}}{2}, - \frac{3 + \sqrt{29}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 6, - \frac{3 - \sqrt{29}}{2}, - \frac{3 + \sqrt{29}}{2}$

Decimal Form:

$x = - 6, 1.19258240 \dots, - 4.19258240 \dots$