Find All Complex Solutions x^3-3x^2+x=-5

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

Step 1

Add $5$ to both sides of the equation.

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

Step 2

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

$p = \pm 1, \pm 5 q = \pm 1$

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

$\pm 1, \pm 5$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 3 {(- 1)}^{2} - 1 + 5$

Raise $- 1$ to the power of $3$ .

$- 1 - 3 {(- 1)}^{2} - 1 + 5$

Raise $- 1$ to the power of $2$ .

$- 1 - 3 \cdot 1 - 1 + 5$

Multiply $- 3$ by $1$ .

$- 1 - 3 - 1 + 5$

Subtract $3$ from $- 1$ .

$- 4 - 1 + 5$

Subtract $1$ from $- 4$ .

$- 5 + 5$

Add $- 5$ and $5$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 3 x^{2} + x + 5}{x + 1}$

Divide $x^{3} - 3 x^{2} + x + 5$ by $x + 1$ .

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$x$

$5$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					-	$4 x^{2}$	-	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 4 x$

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$
							+	$5 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$
							+	$5 x$	+	$5$

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

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$
							+	$5 x$	+	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$
							+	$5 x$	+	$5$
							+	$5 x$	+	$5$

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

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$
							+	$5 x$	+	$5$
							-	$5 x$	-	$5$

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

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$x$	+	$5$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$x$
					+	$4 x^{2}$	+	$4 x$
							+	$5 x$	+	$5$
							-	$5 x$	-	$5$
										$0$

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

$x^{2} - 4 x + 5$

Write $x^{3} - 3 x^{2} + x + 5$ as a set of factors.

$(x + 1) (x^{2} - 4 x + 5) = 0$

Step 3

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

$x + 1 = 0$

$x^{2} - 4 x + 5 = 0$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x^{2} - 4 x + 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 4 x + 5$ equal to $0$ .

$x^{2} - 4 x + 5 = 0$

Solve $x^{2} - 4 x + 5 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Rewrite $\sqrt{- 1}$ as $i$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Rewrite $\sqrt{- 1}$ as $i$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Change the $\pm$ to $+$ .

$x = 2 + i$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Rewrite $\sqrt{- 1}$ as $i$ .

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

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Change the $\pm$ to $-$ .

$x = 2 - i$

The final answer is the combination of both solutions.

$x = 2 + i, 2 - i$

Step 6

The final solution is all the values that make $(x + 1) (x^{2} - 4 x + 5) = 0$ true.

$x = - 1, 2 + i, 2 - i$

Find All Complex Solutions x^3-3x^2+x=-5

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