Find the Roots (Zeros) x^3-4x^2+x+26=0

$x^{3} - 4 x^{2} + x + 26 = 0$

Step 1

Factor $x^{3} - 4 x^{2} + x + 26$ 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.

$p = \pm 1, \pm 26, \pm 2, \pm 13 q = \pm 1$

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

$\pm 1, \pm 26, \pm 2, \pm 13$

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 4 {(- 2)}^{2} - 2 + 26$

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

$- 8 - 4 {(- 2)}^{2} - 2 + 26$

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

$- 8 - 4 \cdot 4 - 2 + 26$

Multiply $- 4$ by $4$ .

$- 8 - 16 - 2 + 26$

Subtract $16$ from $- 8$ .

$- 24 - 2 + 26$

Subtract $2$ from $- 24$ .

$- 26 + 26$

Add $- 26$ and $26$ .

$0$

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 4 x^{2} + x + 26}{x + 2}$

Divide $x^{3} - 4 x^{2} + x + 26$ by $x + 2$ .

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$x$

$26$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					-	$6 x^{2}$	-	$12 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 12 x$

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$
							+	$13 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$
							+	$13 x$	+	$26$

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$13$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$
							+	$13 x$	+	$26$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$13$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$
							+	$13 x$	+	$26$
							+	$13 x$	+	$26$

The expression needs to be subtracted from the dividend, so change all the signs in $13 x + 26$

				$x^{2}$	-	$6 x$	+	$13$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$
							+	$13 x$	+	$26$
							-	$13 x$	-	$26$

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

				$x^{2}$	-	$6 x$	+	$13$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$26$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	+	$x$
					+	$6 x^{2}$	+	$12 x$
							+	$13 x$	+	$26$
							-	$13 x$	-	$26$
										$0$

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

$x^{2} - 6 x + 13$

Write $x^{3} - 4 x^{2} + x + 26$ as a set of factors.

$(x + 2) (x^{2} - 6 x + 13) = 0$

Step 2

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

$x + 2 = 0$

$x^{2} - 6 x + 13 = 0$

Step 3

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

Set $x^{2} - 6 x + 13$ equal to $0$ and solve for $x$ .

Set $x^{2} - 6 x + 13$ equal to $0$ .

$x^{2} - 6 x + 13 = 0$

Solve $x^{2} - 6 x + 13 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 6$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 i}{2}$

Simplify $\frac{6 \pm 4 i}{2}$ .

$x = 3 \pm 2 i$

The final answer is the combination of both solutions.

$x = 3 + 2 i, 3 - 2 i$

Step 5

The final solution is all the values that make $(x + 2) (x^{2} - 6 x + 13) = 0$ true.

$x = - 2, 3 + 2 i, 3 - 2 i$

Step 6

Find the Roots (Zeros) x^3-4x^2+x+26=0

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