Solve by Factoring x^3+9x^2+8x-60=(x+6)

$x^{3} + 9 x^{2} + 8 x - 60 = (x + 6)$

Step 1

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

$x^{3} + 9 x^{2} + 8 x - 60 - (x + 6) = 0$

Step 2

Simplify $x^{3} + 9 x^{2} + 8 x - 60 - (x + 6)$ .

Simplify each term.

Apply the distributive property.

$x^{3} + 9 x^{2} + 8 x - 60 - x - 1 \cdot 6 = 0$

Multiply $- 1$ by $6$ .

$x^{3} + 9 x^{2} + 8 x - 60 - x - 6 = 0$

Subtract $x$ from $8 x$ .

$x^{3} + 9 x^{2} + 7 x - 60 - 6 = 0$

Subtract $6$ from $- 60$ .

$x^{3} + 9 x^{2} + 7 x - 66 = 0$

Step 3

Factor $x^{3} + 9 x^{2} + 7 x - 66$ using the rational roots test.

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.

$p = \pm 1, \pm 66, \pm 2, \pm 33, \pm 3, \pm 22, \pm 6, \pm 11 q = \pm 1$

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

$\pm 1, \pm 66, \pm 2, \pm 33, \pm 3, \pm 22, \pm 6, \pm 11$

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

Substitute $- 6$ into the polynomial.

${(- 6)}^{3} + 9 {(- 6)}^{2} + 7 \cdot - 6 - 66$

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

$- 216 + 9 {(- 6)}^{2} + 7 \cdot - 6 - 66$

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

$- 216 + 9 \cdot 36 + 7 \cdot - 6 - 66$

Multiply $9$ by $36$ .

$- 216 + 324 + 7 \cdot - 6 - 66$

Add $- 216$ and $324$ .

$108 + 7 \cdot - 6 - 66$

Multiply $7$ by $- 6$ .

$108 - 42 - 66$

Subtract $42$ from $108$ .

$66 - 66$

Subtract $66$ from $66$ .

$0$

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.

$\frac{x^{3} + 9 x^{2} + 7 x - 66}{x + 6}$

Divide $x^{3} + 9 x^{2} + 7 x - 66$ by $x + 6$ .

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

$x$

$6$

$x^{3}$

$9 x^{2}$

$7 x$

$66$

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

					$x^{2}$
	$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			+	$x^{3}$	+	$6 x^{2}$

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					+	$3 x^{2}$	+	$18 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$
							-	$11 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$
							-	$11 x$	-	$66$

Divide the highest order term in the dividend $- 11 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$11$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$
							-	$11 x$	-	$66$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$11$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$
							-	$11 x$	-	$66$
							-	$11 x$	-	$66$

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x - 66$

				$x^{2}$	+	$3 x$	-	$11$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$
							-	$11 x$	-	$66$
							+	$11 x$	+	$66$

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

				$x^{2}$	+	$3 x$	-	$11$
$x$	+	$6$		$x^{3}$	+	$9 x^{2}$	+	$7 x$	-	$66$
			-	$x^{3}$	-	$6 x^{2}$
					+	$3 x^{2}$	+	$7 x$
					-	$3 x^{2}$	-	$18 x$
							-	$11 x$	-	$66$
							+	$11 x$	+	$66$
										$0$

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

$x^{2} + 3 x - 11$

Write $x^{3} + 9 x^{2} + 7 x - 66$ as a set of factors.

$(x + 6) (x^{2} + 3 x - 11) = 0$

Step 4

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

$x + 6 = 0$

$x^{2} + 3 x - 11 = 0$

Step 5

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 6

Set $x^{2} + 3 x - 11$ equal to $0$ and solve for $x$ .

Set $x^{2} + 3 x - 11$ equal to $0$ .

$x^{2} + 3 x - 11 = 0$

Solve $x^{2} + 3 x - 11 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 3$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$x = \frac{- 3 \pm \sqrt{9 + 44}}{2 \cdot 1}$

Add $9$ and $44$ .

$x = \frac{- 3 \pm \sqrt{53}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{53}}{2}$

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

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$x = \frac{- 3 \pm \sqrt{9 + 44}}{2 \cdot 1}$

Add $9$ and $44$ .

$x = \frac{- 3 \pm \sqrt{53}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{53}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{53}}{2}$

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

$x = \frac{- 1 \cdot 3 + \sqrt{53}}{2}$

Factor $- 1$ out of $\sqrt{53}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{53})}{2}$

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{53})$ .

$x = \frac{- 1 (3 - \sqrt{53})}{2}$

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{53}}{2}$

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

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$x = \frac{- 3 \pm \sqrt{9 + 44}}{2 \cdot 1}$

Add $9$ and $44$ .

$x = \frac{- 3 \pm \sqrt{53}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{53}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{53}}{2}$

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

$x = \frac{- 1 \cdot 3 - \sqrt{53}}{2}$

Factor $- 1$ out of $- \sqrt{53}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{53})}{2}$

Factor $- 1$ out of $- 1 (3) - (\sqrt{53})$ .

$x = \frac{- 1 (3 + \sqrt{53})}{2}$

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{53}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{53}}{2}, - \frac{3 + \sqrt{53}}{2}$

Step 7

The final solution is all the values that make $(x + 6) (x^{2} + 3 x - 11) = 0$ true.

$x = - 6, - \frac{3 - \sqrt{53}}{2}, - \frac{3 + \sqrt{53}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 6, - \frac{3 - \sqrt{53}}{2}, - \frac{3 + \sqrt{53}}{2}$

Decimal Form:

$x = - 6, 2.14005494 \dots, - 5.14005494 \dots$

Solve by Factoring x^3+9x^2+8x-60=(x+6)

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