Identify the Zeros and Their Multiplicities f(x)=x^3+3x^2-185x-187

$f (x) = x^{3} + 3 x^{2} - 185 x - 187$

Step 1

Set $x^{3} + 3 x^{2} - 185 x - 187$ equal to $0$ .

$x^{3} + 3 x^{2} - 185 x - 187 = 0$

Step 2

Solve for $x$ .

Factor $x^{3} + 3 x^{2} - 185 x - 187$ 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.

$p = \pm 1, \pm 187, \pm 11, \pm 17 q = \pm 1$

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

$\pm 1, \pm 187, \pm 11, \pm 17$

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.

${(- 1)}^{3} + 3 {(- 1)}^{2} - 185 \cdot - 1 - 187$

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

$- 1 + 3 {(- 1)}^{2} - 185 \cdot - 1 - 187$

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

$- 1 + 3 \cdot 1 - 185 \cdot - 1 - 187$

Multiply $3$ by $1$ .

$- 1 + 3 - 185 \cdot - 1 - 187$

Add $- 1$ and $3$ .

$2 - 185 \cdot - 1 - 187$

Multiply $- 185$ by $- 1$ .

$2 + 185 - 187$

Add $2$ and $185$ .

$187 - 187$

Subtract $187$ from $187$ .

$0$

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.

$\frac{x^{3} + 3 x^{2} - 185 x - 187}{x + 1}$

Divide $x^{3} + 3 x^{2} - 185 x - 187$ 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 $0$ .

$x$

$1$

$x^{3}$

$3 x^{2}$

$185 x$

$187$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					+	$2 x^{2}$	+	$2 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$
							-	$187 x$

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

				$x^{2}$	+	$2 x$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$
							-	$187 x$	-	$187$

Divide the highest order term in the dividend $- 187 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$2 x$	-	$187$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$
							-	$187 x$	-	$187$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$187$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$
							-	$187 x$	-	$187$
							-	$187 x$	-	$187$

The expression needs to be subtracted from the dividend, so change all the signs in $- 187 x - 187$

				$x^{2}$	+	$2 x$	-	$187$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$
							-	$187 x$	-	$187$
							+	$187 x$	+	$187$

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

				$x^{2}$	+	$2 x$	-	$187$
$x$	+	$1$		$x^{3}$	+	$3 x^{2}$	-	$185 x$	-	$187$
			-	$x^{3}$	-	$x^{2}$
					+	$2 x^{2}$	-	$185 x$
					-	$2 x^{2}$	-	$2 x$
							-	$187 x$	-	$187$
							+	$187 x$	+	$187$
										$0$

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

$x^{2} + 2 x - 187$

Write $x^{3} + 3 x^{2} - 185 x - 187$ as a set of factors.

$(x + 1) (x^{2} + 2 x - 187) = 0$

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

$x + 1 = 0$

$x^{2} + 2 x - 187 = 0$

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.

$x = - 1$

Set $x^{2} + 2 x - 187$ equal to $0$ and solve for $x$ .

Set $x^{2} + 2 x - 187$ equal to $0$ .

$x^{2} + 2 x - 187 = 0$

Solve $x^{2} + 2 x - 187 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 2$ , and $c = - 187$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 187)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 187}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 187$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 187}}{2 \cdot 1}$

Multiply $- 4$ by $- 187$ .

$x = \frac{- 2 \pm \sqrt{4 + 748}}{2 \cdot 1}$

Add $4$ and $748$ .

$x = \frac{- 2 \pm \sqrt{752}}{2 \cdot 1}$

Rewrite $752$ as $4^{2} \cdot 47$ .

Factor $16$ out of $752$ .

$x = \frac{- 2 \pm \sqrt{16 (47)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{4^{2} \cdot 47}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 4 \sqrt{47}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 4 \sqrt{47}}{2}$

Simplify $\frac{- 2 \pm 4 \sqrt{47}}{2}$ .

$x = - 1 \pm 2 \sqrt{47}$

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

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 187}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 187$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 187}}{2 \cdot 1}$

Multiply $- 4$ by $- 187$ .

$x = \frac{- 2 \pm \sqrt{4 + 748}}{2 \cdot 1}$

Add $4$ and $748$ .

$x = \frac{- 2 \pm \sqrt{752}}{2 \cdot 1}$

Rewrite $752$ as $4^{2} \cdot 47$ .

Factor $16$ out of $752$ .

$x = \frac{- 2 \pm \sqrt{16 (47)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{4^{2} \cdot 47}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 4 \sqrt{47}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 4 \sqrt{47}}{2}$

Simplify $\frac{- 2 \pm 4 \sqrt{47}}{2}$ .

$x = - 1 \pm 2 \sqrt{47}$

Change the $\pm$ to $+$ .

$x = - 1 + 2 \sqrt{47}$

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

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 187}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 187$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 187}}{2 \cdot 1}$

Multiply $- 4$ by $- 187$ .

$x = \frac{- 2 \pm \sqrt{4 + 748}}{2 \cdot 1}$

Add $4$ and $748$ .

$x = \frac{- 2 \pm \sqrt{752}}{2 \cdot 1}$

Rewrite $752$ as $4^{2} \cdot 47$ .

Factor $16$ out of $752$ .

$x = \frac{- 2 \pm \sqrt{16 (47)}}{2 \cdot 1}$

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

$x = \frac{- 2 \pm \sqrt{4^{2} \cdot 47}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 4 \sqrt{47}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 4 \sqrt{47}}{2}$

Simplify $\frac{- 2 \pm 4 \sqrt{47}}{2}$ .

$x = - 1 \pm 2 \sqrt{47}$

Change the $\pm$ to $-$ .

$x = - 1 - 2 \sqrt{47}$

The final answer is the combination of both solutions.

$x = - 1 + 2 \sqrt{47}, - 1 - 2 \sqrt{47}$

The final solution is all the values that make $(x + 1) (x^{2} + 2 x - 187) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 1$ (Multiplicity of $1$ )

$x = - 1 + 2 \sqrt{47}$ (Multiplicity of $1$ )

$x = - 1 - 2 \sqrt{47}$ (Multiplicity of $1$ )

$x = - 1$ (Multiplicity of $1$ )

$x = - 1 + 2 \sqrt{47}$ (Multiplicity of $1$ )

$x = - 1 - 2 \sqrt{47}$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=x^3+3x^2-185x-187

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