Identify the Zeros and Their Multiplicities x^3+4x^2-218x-221

$x^{3} + 4 x^{2} - 218 x - 221$

Step 1

Set $x^{3} + 4 x^{2} - 218 x - 221$ equal to $0$ .

$x^{3} + 4 x^{2} - 218 x - 221 = 0$

Step 2

Solve for $x$ .

Factor $x^{3} + 4 x^{2} - 218 x - 221$ 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.

$p = \pm 1, \pm 221, \pm 13, \pm 17 q = \pm 1$

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

$\pm 1, \pm 221, \pm 13, \pm 17$

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.

${(- 1)}^{3} + 4 {(- 1)}^{2} - 218 \cdot - 1 - 221$

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

$- 1 + 4 {(- 1)}^{2} - 218 \cdot - 1 - 221$

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

$- 1 + 4 \cdot 1 - 218 \cdot - 1 - 221$

Multiply $4$ by $1$ .

$- 1 + 4 - 218 \cdot - 1 - 221$

Add $- 1$ and $4$ .

$3 - 218 \cdot - 1 - 221$

Multiply $- 218$ by $- 1$ .

$3 + 218 - 221$

Add $3$ and $218$ .

$221 - 221$

Subtract $221$ from $221$ .

$0$

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.

$\frac{x^{3} + 4 x^{2} - 218 x - 221}{x + 1}$

Divide $x^{3} + 4 x^{2} - 218 x - 221$ 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 $0$ .

$x$

$1$

$x^{3}$

$4 x^{2}$

$218 x$

$221$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					+	$3 x^{2}$	+	$3 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$
							-	$221 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$
							-	$221 x$	-	$221$

Divide the highest order term in the dividend $- 221 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$221$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$
							-	$221 x$	-	$221$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$221$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$
							-	$221 x$	-	$221$
							-	$221 x$	-	$221$

The expression needs to be subtracted from the dividend, so change all the signs in $- 221 x - 221$

				$x^{2}$	+	$3 x$	-	$221$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$
							-	$221 x$	-	$221$
							+	$221 x$	+	$221$

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

				$x^{2}$	+	$3 x$	-	$221$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$218 x$	-	$221$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$218 x$
					-	$3 x^{2}$	-	$3 x$
							-	$221 x$	-	$221$
							+	$221 x$	+	$221$
										$0$

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

$x^{2} + 3 x - 221$

Write $x^{3} + 4 x^{2} - 218 x - 221$ as a set of factors.

$(x + 1) (x^{2} + 3 x - 221) = 0$

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

$x + 1 = 0$

$x^{2} + 3 x - 221 = 0$

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 $x^{2} + 3 x - 221$ equal to $0$ and solve for $x$ .

Set $x^{2} + 3 x - 221$ equal to $0$ .

$x^{2} + 3 x - 221 = 0$

Solve $x^{2} + 3 x - 221 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 3$ , and $c = - 221$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 221)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 221}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 221$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 221}}{2 \cdot 1}$

Multiply $- 4$ by $- 221$ .

$x = \frac{- 3 \pm \sqrt{9 + 884}}{2 \cdot 1}$

Add $9$ and $884$ .

$x = \frac{- 3 \pm \sqrt{893}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{893}}{2}$

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

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 221}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 221$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 221}}{2 \cdot 1}$

Multiply $- 4$ by $- 221$ .

$x = \frac{- 3 \pm \sqrt{9 + 884}}{2 \cdot 1}$

Add $9$ and $884$ .

$x = \frac{- 3 \pm \sqrt{893}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{893}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{893}}{2}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{893}}{2}$

Factor $- 1$ out of $\sqrt{893}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{893})}{2}$

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{893})$ .

$x = \frac{- 1 (3 - \sqrt{893})}{2}$

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{893}}{2}$

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

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 221}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 221$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 221}}{2 \cdot 1}$

Multiply $- 4$ by $- 221$ .

$x = \frac{- 3 \pm \sqrt{9 + 884}}{2 \cdot 1}$

Add $9$ and $884$ .

$x = \frac{- 3 \pm \sqrt{893}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{893}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{893}}{2}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{893}}{2}$

Factor $- 1$ out of $- \sqrt{893}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{893})}{2}$

Factor $- 1$ out of $- 1 (3) - (\sqrt{893})$ .

$x = \frac{- 1 (3 + \sqrt{893})}{2}$

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{893}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{893}}{2}, - \frac{3 + \sqrt{893}}{2}$

The final solution is all the values that make $(x + 1) (x^{2} + 3 x - 221) = 0$ true. The multiplicity of a root is the number of times the root appears.

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

$x = - \frac{3 - \sqrt{893}}{2}$ (Multiplicity of $1$ )

$x = - \frac{3 + \sqrt{893}}{2}$ (Multiplicity of $1$ )

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

$x = - \frac{3 - \sqrt{893}}{2}$ (Multiplicity of $1$ )

$x = - \frac{3 + \sqrt{893}}{2}$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities x^3+4x^2-218x-221

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