Solve by Factoring x^3+7x^2+2x-40=(x+5)

$x^{3} + 7 x^{2} + 2 x - 40 = (x + 5)$

Step 1

Subtract $(x + 5)$ from both sides of the equation.

$x^{3} + 7 x^{2} + 2 x - 40 - (x + 5) = 0$

Step 2

Simplify $x^{3} + 7 x^{2} + 2 x - 40 - (x + 5)$ .

Simplify each term.

Apply the distributive property.

$x^{3} + 7 x^{2} + 2 x - 40 - x - 1 \cdot 5 = 0$

Multiply $- 1$ by $5$ .

$x^{3} + 7 x^{2} + 2 x - 40 - x - 5 = 0$

Subtract $x$ from $2 x$ .

$x^{3} + 7 x^{2} + x - 40 - 5 = 0$

Subtract $5$ from $- 40$ .

$x^{3} + 7 x^{2} + x - 45 = 0$

Step 3

Factor $x^{3} + 7 x^{2} + x - 45$ 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.

$p = \pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9 q = \pm 1$

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

$\pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9$

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

Substitute $- 5$ into the polynomial.

${(- 5)}^{3} + 7 {(- 5)}^{2} - 5 - 45$

Raise $- 5$ to the power of $3$ .

$- 125 + 7 {(- 5)}^{2} - 5 - 45$

Raise $- 5$ to the power of $2$ .

$- 125 + 7 \cdot 25 - 5 - 45$

Multiply $7$ by $25$ .

$- 125 + 175 - 5 - 45$

Add $- 125$ and $175$ .

$50 - 5 - 45$

Subtract $5$ from $50$ .

$45 - 45$

Subtract $45$ from $45$ .

$0$

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 7 x^{2} + x - 45}{x + 5}$

Divide $x^{3} + 7 x^{2} + x - 45$ by $x + 5$ .

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

$x$

$5$

$x^{3}$

$7 x^{2}$

$x$

$45$

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

					$x^{2}$
	$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			+	$x^{3}$	+	$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 5 x^{2}$

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$

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

				$x^{2}$	+	$2 x$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					+	$2 x^{2}$	+	$10 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 10 x$

				$x^{2}$	+	$2 x$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$
							-	$9 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$
							-	$9 x$	-	$45$

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

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$
							-	$9 x$	-	$45$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$
							-	$9 x$	-	$45$
							-	$9 x$	-	$45$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x - 45$

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$
							-	$9 x$	-	$45$
							+	$9 x$	+	$45$

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

				$x^{2}$	+	$2 x$	-	$9$
$x$	+	$5$		$x^{3}$	+	$7 x^{2}$	+	$x$	-	$45$
			-	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	+	$x$
					-	$2 x^{2}$	-	$10 x$
							-	$9 x$	-	$45$
							+	$9 x$	+	$45$
										$0$

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

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

Write $x^{3} + 7 x^{2} + x - 45$ as a set of factors.

$(x + 5) (x^{2} + 2 x - 9) = 0$

Step 4

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

$x + 5 = 0$

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

Step 5

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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 $(x + 5) (x^{2} + 2 x - 9) = 0$ true.

$x = - 5, - 1 + \sqrt{10}, - 1 - \sqrt{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 5, - 1 + \sqrt{10}, - 1 - \sqrt{10}$

Decimal Form:

$x = - 5, 2.16227766 \dots, - 4.16227766 \dots$

Solve by Factoring x^3+7x^2+2x-40=(x+5)

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