Solve by Factoring x^3+7x^2+4x-12=(x+6)

$x^{3} + 7 x^{2} + 4 x - 12 = (x + 6)$

Step 1

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

$x^{3} + 7 x^{2} + 4 x - 12 - (x + 6) = 0$

Step 2

Simplify $x^{3} + 7 x^{2} + 4 x - 12 - (x + 6)$ .

Simplify each term.

Apply the distributive property.

$x^{3} + 7 x^{2} + 4 x - 12 - x - 1 \cdot 6 = 0$

Multiply $- 1$ by $6$ .

$x^{3} + 7 x^{2} + 4 x - 12 - x - 6 = 0$

Subtract $x$ from $4 x$ .

$x^{3} + 7 x^{2} + 3 x - 12 - 6 = 0$

Subtract $6$ from $- 12$ .

$x^{3} + 7 x^{2} + 3 x - 18 = 0$

Step 3

Factor $x^{3} + 7 x^{2} + 3 x - 18$ 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.

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

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

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

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.

${(- 6)}^{3} + 7 {(- 6)}^{2} + 3 \cdot - 6 - 18$

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

$- 216 + 7 {(- 6)}^{2} + 3 \cdot - 6 - 18$

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

$- 216 + 7 \cdot 36 + 3 \cdot - 6 - 18$

Multiply $7$ by $36$ .

$- 216 + 252 + 3 \cdot - 6 - 18$

Add $- 216$ and $252$ .

$36 + 3 \cdot - 6 - 18$

Multiply $3$ by $- 6$ .

$36 - 18 - 18$

Subtract $18$ from $36$ .

$18 - 18$

Subtract $18$ from $18$ .

$0$

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.

$\frac{x^{3} + 7 x^{2} + 3 x - 18}{x + 6}$

Divide $x^{3} + 7 x^{2} + 3 x - 18$ 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 $0$ .

$x$

$6$

$x^{3}$

$7 x^{2}$

$3 x$

$18$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x - 18$

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

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

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

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

$x^{2} + x - 3$

Write $x^{3} + 7 x^{2} + 3 x - 18$ as a set of factors.

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

Step 4

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

$x + 6 = 0$

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

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

One to any power is one.

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

Multiply $- 4 \cdot 1 \cdot - 3$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 12}}{2 \cdot 1}$

Add $1$ and $12$ .

$x = \frac{- 1 \pm \sqrt{13}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{13}}{2}$

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

Simplify the numerator.

One to any power is one.

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

Multiply $- 4 \cdot 1 \cdot - 3$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 12}}{2 \cdot 1}$

Add $1$ and $12$ .

$x = \frac{- 1 \pm \sqrt{13}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{13}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{13}}{2}$

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

$x = \frac{- 1 \cdot 1 + \sqrt{13}}{2}$

Factor $- 1$ out of $\sqrt{13}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{13})}{2}$

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{13})$ .

$x = \frac{- 1 (1 - \sqrt{13})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{13}}{2}$

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

Simplify the numerator.

One to any power is one.

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

Multiply $- 4 \cdot 1 \cdot - 3$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 12}}{2 \cdot 1}$

Add $1$ and $12$ .

$x = \frac{- 1 \pm \sqrt{13}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{13}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{13}}{2}$

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

$x = \frac{- 1 \cdot 1 - \sqrt{13}}{2}$

Factor $- 1$ out of $- \sqrt{13}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{13})}{2}$

Factor $- 1$ out of $- 1 (1) - (\sqrt{13})$ .

$x = \frac{- 1 (1 + \sqrt{13})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{13}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{13}}{2}, - \frac{1 + \sqrt{13}}{2}$

Step 7

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

$x = - 6, - \frac{1 - \sqrt{13}}{2}, - \frac{1 + \sqrt{13}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 6, - \frac{1 - \sqrt{13}}{2}, - \frac{1 + \sqrt{13}}{2}$

Decimal Form:

$x = - 6, 1.30277563 \dots, - 2.30277563 \dots$

Solve by Factoring x^3+7x^2+4x-12=(x+6)

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