Find the Roots (Zeros) f(x)=x^4-16x^3+50x^2+152x+85

$f (x) = x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85$

Step 1

Set $x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85$ equal to $0$ .

$x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85$ 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.

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

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

$\pm 1, \pm 85, \pm 5, \pm 17$

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.

${(- 1)}^{4} - 16 {(- 1)}^{3} + 50 {(- 1)}^{2} + 152 \cdot - 1 + 85$

Raise $- 1$ to the power of $4$ .

$1 - 16 {(- 1)}^{3} + 50 {(- 1)}^{2} + 152 \cdot - 1 + 85$

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

$1 - 16 \cdot - 1 + 50 {(- 1)}^{2} + 152 \cdot - 1 + 85$

Multiply $- 16$ by $- 1$ .

$1 + 16 + 50 {(- 1)}^{2} + 152 \cdot - 1 + 85$

Add $1$ and $16$ .

$17 + 50 {(- 1)}^{2} + 152 \cdot - 1 + 85$

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

$17 + 50 \cdot 1 + 152 \cdot - 1 + 85$

Multiply $50$ by $1$ .

$17 + 50 + 152 \cdot - 1 + 85$

Add $17$ and $50$ .

$67 + 152 \cdot - 1 + 85$

Multiply $152$ by $- 1$ .

$67 - 152 + 85$

Subtract $152$ from $67$ .

$- 85 + 85$

Add $- 85$ and $85$ .

$0$

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.

$\frac{x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85}{x + 1}$

Divide $x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85$ 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 $0$ .

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

Divide the highest order term in the dividend $- 17 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$17 x^{2}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$17 x^{2}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 17 x^{3} - 17 x^{2}$

$x^{3}$

$17 x^{2}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

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

$x^{3}$

$17 x^{2}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

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

$x^{3}$

$17 x^{2}$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

Divide the highest order term in the dividend $67 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$17 x^{2}$

$67 x$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$17 x^{2}$

$67 x$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

The expression needs to be subtracted from the dividend, so change all the signs in $67 x^{2} + 67 x$

$x^{3}$

$17 x^{2}$

$67 x$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

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

$x^{3}$

$17 x^{2}$

$67 x$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

$85 x$

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

$x^{3}$

$17 x^{2}$

$67 x$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

$85 x$

$85$

Divide the highest order term in the dividend $85 x$ by the highest order term in divisor $x$ .

$x^{3}$

$17 x^{2}$

$67 x$

$85$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

$85 x$

$85$

Multiply the new quotient term by the divisor.

$x^{3}$

$17 x^{2}$

$67 x$

$85$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

$85 x$

$85$

$85 x$

$85$

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

$x^{3}$

$17 x^{2}$

$67 x$

$85$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

$85 x$

$85$

$85 x$

$85$

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

$x^{3}$

$17 x^{2}$

$67 x$

$85$

$x$

$1$

$x^{4}$

$16 x^{3}$

$50 x^{2}$

$152 x$

$85$

$x^{4}$

$x^{3}$

$17 x^{3}$

$50 x^{2}$

$17 x^{3}$

$17 x^{2}$

$67 x^{2}$

$152 x$

$67 x^{2}$

$67 x$

$85 x$

$85$

$85 x$

$85$

$0$

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

$x^{3} - 17 x^{2} + 67 x + 85$

Write $x^{4} - 16 x^{3} + 50 x^{2} + 152 x + 85$ as a set of factors.

$(x + 1) (x^{3} - 17 x^{2} + 67 x + 85) = 0$

Factor $x^{3} - 17 x^{2} + 67 x + 85$ 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.

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

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

$\pm 1, \pm 85, \pm 5, \pm 17$

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.

${(- 1)}^{3} - 17 {(- 1)}^{2} + 67 \cdot - 1 + 85$

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

$- 1 - 17 {(- 1)}^{2} + 67 \cdot - 1 + 85$

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

$- 1 - 17 \cdot 1 + 67 \cdot - 1 + 85$

Multiply $- 17$ by $1$ .

$- 1 - 17 + 67 \cdot - 1 + 85$

Subtract $17$ from $- 1$ .

$- 18 + 67 \cdot - 1 + 85$

Multiply $67$ by $- 1$ .

$- 18 - 67 + 85$

Subtract $67$ from $- 18$ .

$- 85 + 85$

Add $- 85$ and $85$ .

$0$

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.

$\frac{x^{3} - 17 x^{2} + 67 x + 85}{x + 1}$

Divide $x^{3} - 17 x^{2} + 67 x + 85$ 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 $0$ .

$x$

$1$

$x^{3}$

$17 x^{2}$

$67 x$

$85$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$

Divide the highest order term in the dividend $- 18 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$18 x$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$18 x$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					-	$18 x^{2}$	-	$18 x$

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

				$x^{2}$	-	$18 x$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$

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

				$x^{2}$	-	$18 x$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$
							+	$85 x$

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

				$x^{2}$	-	$18 x$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$
							+	$85 x$	+	$85$

Divide the highest order term in the dividend $85 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$18 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$
							+	$85 x$	+	$85$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$18 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$
							+	$85 x$	+	$85$
							+	$85 x$	+	$85$

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

				$x^{2}$	-	$18 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$
							+	$85 x$	+	$85$
							-	$85 x$	-	$85$

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

				$x^{2}$	-	$18 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$17 x^{2}$	+	$67 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$18 x^{2}$	+	$67 x$
					+	$18 x^{2}$	+	$18 x$
							+	$85 x$	+	$85$
							-	$85 x$	-	$85$
										$0$

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

$x^{2} - 18 x + 85$

Write $x^{3} - 17 x^{2} + 67 x + 85$ as a set of factors.

$(x + 1) ((x + 1) (x^{2} - 18 x + 85)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x + 1) (x^{2} - 18 x + 85) = 0$

Combine like factors.

Raise $x + 1$ to the power of $1$ .

$(x + 1) (x + 1) (x^{2} - 18 x + 85) = 0$

Raise $x + 1$ to the power of $1$ .

$(x + 1) (x + 1) (x^{2} - 18 x + 85) = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x + 1)}^{1 + 1} (x^{2} - 18 x + 85) = 0$

Add $1$ and $1$ .

${(x + 1)}^{2} (x^{2} - 18 x + 85) = 0$

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

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

$x^{2} - 18 x + 85 = 0$

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

Set ${(x + 1)}^{2}$ equal to $0$ .

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

Solve ${(x + 1)}^{2} = 0$ for $x$ .

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x^{2} - 18 x + 85$ equal to $0$ and solve for $x$ .

Set $x^{2} - 18 x + 85$ equal to $0$ .

$x^{2} - 18 x + 85 = 0$

Solve $x^{2} - 18 x + 85 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 18$ , and $c = 85$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 85)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 85}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 85}}{2 \cdot 1}$

Multiply $- 4$ by $85$ .

$x = \frac{18 \pm \sqrt{324 - 340}}{2 \cdot 1}$

Subtract $340$ from $324$ .

$x = \frac{18 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{18 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{18 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{18 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

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

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

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

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

Move $4$ to the left of $i$ .

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

Multiply $2$ by $1$ .

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

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

$x = 9 \pm 2 i$

The final answer is the combination of both solutions.

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

The final solution is all the values that make ${(x + 1)}^{2} (x^{2} - 18 x + 85) = 0$ true.

$x = - 1, 9 + 2 i, 9 - 2 i$

Step 3

Find the Roots (Zeros) f(x)=x^4-16x^3+50x^2+152x+85

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