Find the Roots (Zeros) x^3-5x^2+2=-x^3+17x

$x^{3} - 5 x^{2} + 2 = - x^{3} + 17 x$

Step 1

Move all terms containing $x$ to the left side of the equation.

Add $x^{3}$ to both sides of the equation.

$x^{3} - 5 x^{2} + 2 + x^{3} = 17 x$

Subtract $17 x$ from both sides of the equation.

$x^{3} - 5 x^{2} + 2 + x^{3} - 17 x = 0$

Add $x^{3}$ and $x^{3}$ .

$2 x^{3} - 5 x^{2} + 2 - 17 x = 0$

Step 2

Factor the left side of the equation.

Reorder terms.

$2 x^{3} - 5 x^{2} - 17 x + 2 = 0$

Factor $2 x^{3} - 5 x^{2} - 17 x + 2$ 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.

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

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

$\pm 1, \pm 0.5, \pm 2$

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.

$2 {(- 2)}^{3} - 5 {(- 2)}^{2} - 17 \cdot - 2 + 2$

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

$2 \cdot - 8 - 5 {(- 2)}^{2} - 17 \cdot - 2 + 2$

Multiply $2$ by $- 8$ .

$- 16 - 5 {(- 2)}^{2} - 17 \cdot - 2 + 2$

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

$- 16 - 5 \cdot 4 - 17 \cdot - 2 + 2$

Multiply $- 5$ by $4$ .

$- 16 - 20 - 17 \cdot - 2 + 2$

Subtract $20$ from $- 16$ .

$- 36 - 17 \cdot - 2 + 2$

Multiply $- 17$ by $- 2$ .

$- 36 + 34 + 2$

Add $- 36$ and $34$ .

$- 2 + 2$

Add $- 2$ and $2$ .

$0$

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.

$\frac{2 x^{3} - 5 x^{2} - 17 x + 2}{x + 2}$

Divide $2 x^{3} - 5 x^{2} - 17 x + 2$ 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 $0$ .

$x$

$2$

$2 x^{3}$

$5 x^{2}$

$17 x$

$2$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			+	$2 x^{3}$	+	$4 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 4 x^{2}$

				$2 x^{2}$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$

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

				$2 x^{2}$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$

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

				$2 x^{2}$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$

Divide the highest order term in the dividend $- 9 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$9 x$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$9 x$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					-	$9 x^{2}$	-	$18 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{2} - 18 x$

				$2 x^{2}$	-	$9 x$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$

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

				$2 x^{2}$	-	$9 x$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$
							+	$x$

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

				$2 x^{2}$	-	$9 x$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$
							+	$x$	+	$2$

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

				$2 x^{2}$	-	$9 x$	+	$1$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$
							+	$x$	+	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$9 x$	+	$1$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$
							+	$x$	+	$2$
							+	$x$	+	$2$

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

				$2 x^{2}$	-	$9 x$	+	$1$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$
							+	$x$	+	$2$
							-	$x$	-	$2$

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

				$2 x^{2}$	-	$9 x$	+	$1$
$x$	+	$2$		$2 x^{3}$	-	$5 x^{2}$	-	$17 x$	+	$2$
			-	$2 x^{3}$	-	$4 x^{2}$
					-	$9 x^{2}$	-	$17 x$
					+	$9 x^{2}$	+	$18 x$
							+	$x$	+	$2$
							-	$x$	-	$2$
										$0$

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

$2 x^{2} - 9 x + 1$

Write $2 x^{3} - 5 x^{2} - 17 x + 2$ as a set of factors.

$(x + 2) (2 x^{2} - 9 x + 1) = 0$

Step 3

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

$x + 2 = 0$

$2 x^{2} - 9 x + 1 = 0$

Step 4

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 5

Set $2 x^{2} - 9 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x^{2} - 9 x + 1$ equal to $0$ .

$2 x^{2} - 9 x + 1 = 0$

Solve $2 x^{2} - 9 x + 1 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 9$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{9 \pm \sqrt{81 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 8}}{2 \cdot 2}$

Subtract $8$ from $81$ .

$x = \frac{9 \pm \sqrt{73}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{9 \pm \sqrt{73}}{4}$

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{73}}{4}, \frac{9 - \sqrt{73}}{4}$

Step 6

The final solution is all the values that make $(x + 2) (2 x^{2} - 9 x + 1) = 0$ true.

$x = - 2, \frac{9 + \sqrt{73}}{4}, \frac{9 - \sqrt{73}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 2, \frac{9 + \sqrt{73}}{4}, \frac{9 - \sqrt{73}}{4}$

Decimal Form:

$x = - 2, 4.38600093 \dots, 0.11399906 \dots$

Step 8

Find the Roots (Zeros) x^3-5x^2+2=-x^3+17x

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