Identify the Zeros and Their Multiplicities -2x^3-7x^2-166x+85

$- 2 x^{3} - 7 x^{2} - 166 x + 85$

Step 1

Set $- 2 x^{3} - 7 x^{2} - 166 x + 85$ equal to $0$ .

$- 2 x^{3} - 7 x^{2} - 166 x + 85 = 0$

Step 2

Solve for $x$ .

Factor $- 2 x^{3} - 7 x^{2} - 166 x + 85$ 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.

$p = \pm 1, \pm 85, \pm 5, \pm 17 q = \pm 1, \pm 2$

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

$\pm 1, \pm 0.5, \pm 85, \pm 42.5, \pm 5, \pm 2.5, \pm 17, \pm 8.5$

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Substitute $0.5$ into the polynomial.

$- 2 \cdot {0.5}^{3} - 7 \cdot {0.5}^{2} - 166 \cdot 0.5 + 85$

Raise $0.5$ to the power of $3$ .

$- 2 \cdot 0.125 - 7 \cdot {0.5}^{2} - 166 \cdot 0.5 + 85$

Multiply $- 2$ by $0.125$ .

$- 0.25 - 7 \cdot {0.5}^{2} - 166 \cdot 0.5 + 85$

Raise $0.5$ to the power of $2$ .

$- 0.25 - 7 \cdot 0.25 - 166 \cdot 0.5 + 85$

Multiply $- 7$ by $0.25$ .

$- 0.25 - 1.75 - 166 \cdot 0.5 + 85$

Subtract $1.75$ from $- 0.25$ .

$- 2 - 166 \cdot 0.5 + 85$

Multiply $- 166$ by $0.5$ .

$- 2 - 83 + 85$

Subtract $83$ from $- 2$ .

$- 85 + 85$

Add $- 85$ and $85$ .

$0$

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 2 x^{3} - 7 x^{2} - 166 x + 85}{2 x - 1}$

Divide $- 2 x^{3} - 7 x^{2} - 166 x + 85$ by $2 x - 1$ .

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

$2 x$

$1$

$2 x^{3}$

$7 x^{2}$

$166 x$

$85$

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

				-	$x^{2}$
	$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			-	$2 x^{3}$	+	$x^{2}$

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

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$

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

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$

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

			-	$x^{2}$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $2 x$ .

			-	$x^{2}$	-	$4 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$4 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					-	$8 x^{2}$	+	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 4 x$

			-	$x^{2}$	-	$4 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$

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

			-	$x^{2}$	-	$4 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$
							-	$170 x$

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

			-	$x^{2}$	-	$4 x$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$
							-	$170 x$	+	$85$

Divide the highest order term in the dividend $- 170 x$ by the highest order term in divisor $2 x$ .

			-	$x^{2}$	-	$4 x$	-	$85$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$
							-	$170 x$	+	$85$

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$4 x$	-	$85$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$
							-	$170 x$	+	$85$
							-	$170 x$	+	$85$

The expression needs to be subtracted from the dividend, so change all the signs in $- 170 x + 85$

			-	$x^{2}$	-	$4 x$	-	$85$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$
							-	$170 x$	+	$85$
							+	$170 x$	-	$85$

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

			-	$x^{2}$	-	$4 x$	-	$85$
$2 x$	-	$1$	-	$2 x^{3}$	-	$7 x^{2}$	-	$166 x$	+	$85$
			+	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	-	$166 x$
					+	$8 x^{2}$	-	$4 x$
							-	$170 x$	+	$85$
							+	$170 x$	-	$85$
										$0$

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

$- x^{2} - 4 x - 85$

Write $- 2 x^{3} - 7 x^{2} - 166 x + 85$ as a set of factors.

$(2 x - 1) (- x^{2} - 4 x - 85) = 0$

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

$2 x - 1 = 0$

$- x^{2} - 4 x - 85 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Set $- x^{2} - 4 x - 85$ equal to $0$ and solve for $x$ .

Set $- x^{2} - 4 x - 85$ equal to $0$ .

$- x^{2} - 4 x - 85 = 0$

Solve $- x^{2} - 4 x - 85 = 0$ 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 $c = - 85$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot - 85)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 85}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 85$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot - 85}}{2 \cdot - 1}$

Multiply $4$ by $- 85$ .

$x = \frac{4 \pm \sqrt{16 - 340}}{2 \cdot - 1}$

Subtract $340$ from $16$ .

$x = \frac{4 \pm \sqrt{- 324}}{2 \cdot - 1}$

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 324}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot - 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{324}}{2 \cdot - 1}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{18^{2}}}{2 \cdot - 1}$

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

$x = \frac{4 \pm i \cdot 18}{2 \cdot - 1}$

Move $18$ to the left of $i$ .

$x = \frac{4 \pm 18 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 18 i}{- 2}$

Simplify $\frac{4 \pm 18 i}{- 2}$ .

$x = \frac{2 \pm 9 i}{- 1}$

Move the negative one from the denominator of $\frac{2 \pm 9 i}{- 1}$ .

$x = - 1 \cdot (2 \pm 9 i)$

Rewrite $- 1 \cdot (2 \pm 9 i)$ as $- (2 \pm 9 i)$ .

$x = - (2 \pm 9 i)$

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

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 85}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 85$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot - 85}}{2 \cdot - 1}$

Multiply $4$ by $- 85$ .

$x = \frac{4 \pm \sqrt{16 - 340}}{2 \cdot - 1}$

Subtract $340$ from $16$ .

$x = \frac{4 \pm \sqrt{- 324}}{2 \cdot - 1}$

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 324}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot - 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{324}}{2 \cdot - 1}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{18^{2}}}{2 \cdot - 1}$

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

$x = \frac{4 \pm i \cdot 18}{2 \cdot - 1}$

Move $18$ to the left of $i$ .

$x = \frac{4 \pm 18 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 18 i}{- 2}$

Simplify $\frac{4 \pm 18 i}{- 2}$ .

$x = \frac{2 \pm 9 i}{- 1}$

Move the negative one from the denominator of $\frac{2 \pm 9 i}{- 1}$ .

$x = - 1 \cdot (2 \pm 9 i)$

Rewrite $- 1 \cdot (2 \pm 9 i)$ as $- (2 \pm 9 i)$ .

$x = - (2 \pm 9 i)$

Change the $\pm$ to $+$ .

$x = - (2 + 9 i)$

Apply the distributive property.

$x = - 1 \cdot 2 - (9 i)$

Multiply $- 1$ by $2$ .

$x = - 2 - (9 i)$

Multiply $9$ by $- 1$ .

$x = - 2 - 9 i$

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

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 85}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 85$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot - 85}}{2 \cdot - 1}$

Multiply $4$ by $- 85$ .

$x = \frac{4 \pm \sqrt{16 - 340}}{2 \cdot - 1}$

Subtract $340$ from $16$ .

$x = \frac{4 \pm \sqrt{- 324}}{2 \cdot - 1}$

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 324}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot - 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{324}}{2 \cdot - 1}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{18^{2}}}{2 \cdot - 1}$

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

$x = \frac{4 \pm i \cdot 18}{2 \cdot - 1}$

Move $18$ to the left of $i$ .

$x = \frac{4 \pm 18 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 18 i}{- 2}$

Simplify $\frac{4 \pm 18 i}{- 2}$ .

$x = \frac{2 \pm 9 i}{- 1}$

Move the negative one from the denominator of $\frac{2 \pm 9 i}{- 1}$ .

$x = - 1 \cdot (2 \pm 9 i)$

Rewrite $- 1 \cdot (2 \pm 9 i)$ as $- (2 \pm 9 i)$ .

$x = - (2 \pm 9 i)$

Change the $\pm$ to $-$ .

$x = - (2 - 9 i)$

Apply the distributive property.

$x = - 1 \cdot 2 - (- 9 i)$

Multiply $- 1$ by $2$ .

$x = - 2 - (- 9 i)$

Multiply $- 9$ by $- 1$ .

$x = - 2 + 9 i$

The final answer is the combination of both solutions.

$x = - 2 - 9 i, - 2 + 9 i$

The final solution is all the values that make $(2 x - 1) (- x^{2} - 4 x - 85) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = \frac{1}{2}$ (Multiplicity of $1$ )

$x = - 2 - 9 i$ (Multiplicity of $1$ )

$x = - 2 + 9 i$ (Multiplicity of $1$ )

$x = \frac{1}{2}$ (Multiplicity of $1$ )

$x = - 2 - 9 i$ (Multiplicity of $1$ )

$x = - 2 + 9 i$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities -2x^3-7x^2-166x+85

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