Solve by Factoring x^3+8x^2+4x-48=(x+6)

$x^{3} + 8 x^{2} + 4 x - 48 = (x + 6)$

Step 1

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

$x^{3} + 8 x^{2} + 4 x - 48 - (x + 6) = 0$

Step 2

Simplify $x^{3} + 8 x^{2} + 4 x - 48 - (x + 6)$ .

Simplify each term.

Apply the distributive property.

$x^{3} + 8 x^{2} + 4 x - 48 - x - 1 \cdot 6 = 0$

Multiply $- 1$ by $6$ .

$x^{3} + 8 x^{2} + 4 x - 48 - x - 6 = 0$

Subtract $x$ from $4 x$ .

$x^{3} + 8 x^{2} + 3 x - 48 - 6 = 0$

Subtract $6$ from $- 48$ .

$x^{3} + 8 x^{2} + 3 x - 54 = 0$

Step 3

Factor $x^{3} + 8 x^{2} + 3 x - 54$ 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.

$p = \pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9 q = \pm 1$

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

$\pm 1, \pm 54, \pm 2, \pm 27, \pm 3, \pm 18, \pm 6, \pm 9$

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.

${(- 6)}^{3} + 8 {(- 6)}^{2} + 3 \cdot - 6 - 54$

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

$- 216 + 8 {(- 6)}^{2} + 3 \cdot - 6 - 54$

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

$- 216 + 8 \cdot 36 + 3 \cdot - 6 - 54$

Multiply $8$ by $36$ .

$- 216 + 288 + 3 \cdot - 6 - 54$

Add $- 216$ and $288$ .

$72 + 3 \cdot - 6 - 54$

Multiply $3$ by $- 6$ .

$72 - 18 - 54$

Subtract $18$ from $72$ .

$54 - 54$

Subtract $54$ from $54$ .

$0$

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.

$\frac{x^{3} + 8 x^{2} + 3 x - 54}{x + 6}$

Divide $x^{3} + 8 x^{2} + 3 x - 54$ 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 $0$ .

$x$

$6$

$x^{3}$

$8 x^{2}$

$3 x$

$54$

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

					$x^{2}$
	$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			+	$x^{3}$	+	$6 x^{2}$

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	+	$12 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 12 x$

				$x^{2}$	+	$2 x$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$
							-	$9 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$
							-	$9 x$	-	$54$

Divide the highest order term in the dividend $- 9 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$
							-	$9 x$	-	$54$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$
							-	$9 x$	-	$54$
							-	$9 x$	-	$54$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x - 54$

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$
							-	$9 x$	-	$54$
							+	$9 x$	+	$54$

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

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$6$		$x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$54$
			-	$x^{3}$	-	$6 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	-	$12 x$
							-	$9 x$	-	$54$
							+	$9 x$	+	$54$
										$0$

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

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

Write $x^{3} + 8 x^{2} + 3 x - 54$ as a set of factors.

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

Step 4

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

$x + 6 = 0$

$x^{2} + 2 x - 9 = 0$

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 $x^{2} + 2 x - 9$ equal to $0$ and solve for $x$ .

Set $x^{2} + 2 x - 9$ equal to $0$ .

$x^{2} + 2 x - 9 = 0$

Solve $x^{2} + 2 x - 9 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 2$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 9$ .

$x = \frac{- 2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Add $4$ and $36$ .

$x = \frac{- 2 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{- 2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{10}}{2}$ .

$x = - 1 \pm \sqrt{10}$

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

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 9$ .

$x = \frac{- 2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Add $4$ and $36$ .

$x = \frac{- 2 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{- 2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{10}}{2}$ .

$x = - 1 \pm \sqrt{10}$

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{10}$

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

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 9$ .

$x = \frac{- 2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Add $4$ and $36$ .

$x = \frac{- 2 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x = \frac{- 2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 2 \pm 2 \sqrt{10}}{2}$ .

$x = - 1 \pm \sqrt{10}$

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{10}$

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{10}, - 1 - \sqrt{10}$

Step 7

The final solution is all the values that make $(x + 6) (x^{2} + 2 x - 9) = 0$ true.

$x = - 6, - 1 + \sqrt{10}, - 1 - \sqrt{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 6, - 1 + \sqrt{10}, - 1 - \sqrt{10}$

Decimal Form:

$x = - 6, 2.16227766 \dots, - 4.16227766 \dots$

Solve by Factoring x^3+8x^2+4x-48=(x+6)

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