Find the Roots (Zeros) x^3-37x+6=0

$x^{3} - 37 x + 6 = 0$

Step 1

Factor $x^{3} - 37 x + 6$ 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.

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

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

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

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.

$6^{3} - 37 \cdot 6 + 6$

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

$216 - 37 \cdot 6 + 6$

Multiply $- 37$ by $6$ .

$216 - 222 + 6$

Subtract $222$ from $216$ .

$- 6 + 6$

Add $- 6$ and $6$ .

$0$

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.

$\frac{x^{3} - 37 x + 6}{x - 6}$

Divide $x^{3} - 37 x + 6$ 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 $0$ .

$x$

$6$

$x^{3}$

$0 x^{2}$

$37 x$

$6$

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

					$x^{2}$
	$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			+	$x^{3}$	-	$6 x^{2}$

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$

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

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					+	$6 x^{2}$	-	$36 x$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 36 x$

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$

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

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$
							-	$x$

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

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$
							-	$x$	+	$6$

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

				$x^{2}$	+	$6 x$	-	$1$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$
							-	$x$	+	$6$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	-	$1$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$
							-	$x$	+	$6$
							-	$x$	+	$6$

The expression needs to be subtracted from the dividend, so change all the signs in $- x + 6$

				$x^{2}$	+	$6 x$	-	$1$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$
							-	$x$	+	$6$
							+	$x$	-	$6$

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

				$x^{2}$	+	$6 x$	-	$1$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$37 x$	+	$6$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$37 x$
					-	$6 x^{2}$	+	$36 x$
							-	$x$	+	$6$
							+	$x$	-	$6$
										$0$

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

$x^{2} + 6 x - 1$

Write $x^{3} - 37 x + 6$ as a set of factors.

$(x - 6) (x^{2} + 6 x - 1) = 0$

Step 2

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

$x - 6 = 0$

$x^{2} + 6 x - 1 = 0$

Step 3

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Set $x^{2} + 6 x - 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 6 x - 1$ equal to $0$ .

$x^{2} + 6 x - 1 = 0$

Solve $x^{2} + 6 x - 1 = 0$ 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 $c = - 1$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $36$ and $4$ .

$x = \frac{- 6 \pm \sqrt{40}}{2 \cdot 1}$

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

Factor $4$ out of $40$ .

$x = \frac{- 6 \pm \sqrt{4 (10)}}{2 \cdot 1}$

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{10}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{- 6 \pm 2 \sqrt{10}}{2}$ .

$x = - 3 \pm \sqrt{10}$

The final answer is the combination of both solutions.

$x = - 3 + \sqrt{10}, - 3 - \sqrt{10}$

Step 5

The final solution is all the values that make $(x - 6) (x^{2} + 6 x - 1) = 0$ true.

$x = 6, - 3 + \sqrt{10}, - 3 - \sqrt{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 6, - 3 + \sqrt{10}, - 3 - \sqrt{10}$

Decimal Form:

$x = 6, 0.16227766 \dots, - 6.16227766 \dots$

Step 7

Find the Roots (Zeros) x^3-37x+6=0

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