Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-2x^3+78x^2+166x+85

$f (x) = x^{4} - 2 x^{3} + 78 x^{2} + 166 x + 85$

Step 1

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.

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

$q = \pm 1$

Step 2

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

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

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(- 1)}^{4} - 2 {(- 1)}^{3} + 78 {(- 1)}^{2} + 166 (- 1) + 85$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

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

$1 - 2 {(- 1)}^{3} + 78 {(- 1)}^{2} + 166 (- 1) + 85$

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

$1 - 2 \cdot - 1 + 78 {(- 1)}^{2} + 166 (- 1) + 85$

Multiply $- 2$ by $- 1$ .

$1 + 2 + 78 {(- 1)}^{2} + 166 (- 1) + 85$

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

$1 + 2 + 78 \cdot 1 + 166 (- 1) + 85$

Multiply $78$ by $1$ .

$1 + 2 + 78 + 166 (- 1) + 85$

Multiply $166$ by $- 1$ .

$1 + 2 + 78 - 166 + 85$

Simplify by adding and subtracting.

Add $1$ and $2$ .

$3 + 78 - 166 + 85$

Add $3$ and $78$ .

$81 - 166 + 85$

Subtract $166$ from $81$ .

$- 85 + 85$

Add $- 85$ and $85$ .

$0$

Step 5

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.

$\frac{x^{4} - 2 x^{3} + 78 x^{2} + 166 x + 85}{x + 1}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(- 2)$ .

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$			\xa0
	$1$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 1)$ and place the result of $(3)$ under the next term in the dividend $(78)$ .

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$	$3$		\xa0
	$1$	$- 3$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$	$3$		\xa0
	$1$	$- 3$	$81$

Multiply the newest entry in the result $(81)$ by the divisor $(- 1)$ and place the result of $(- 81)$ under the next term in the dividend $(166)$ .

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$	$3$	$- 81$	\xa0
	$1$	$- 3$	$81$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$	$3$	$- 81$	\xa0
	$1$	$- 3$	$81$	$85$

Multiply the newest entry in the result $(85)$ by the divisor $(- 1)$ and place the result of $(- 85)$ under the next term in the dividend $(85)$ .

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$	$3$	$- 81$	$- 85$
	$1$	$- 3$	$81$	$85$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$- 2$	$78$	$166$	$85$
	\xa0	$- 1$	$3$	$- 81$	$- 85$
	$1$	$- 3$	$81$	$85$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 3 x^{2} + (81) x + 85$

Simplify the quotient polynomial.

$x^{3} - 3 x^{2} + 81 x + 85$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 3 x^{2} + 81 x + 85$ 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.

${(- 1)}^{3} - 3 {(- 1)}^{2} + 81 \cdot - 1 + 85$

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

$- 1 - 3 {(- 1)}^{2} + 81 \cdot - 1 + 85$

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

$- 1 - 3 \cdot 1 + 81 \cdot - 1 + 85$

Multiply $- 3$ by $1$ .

$- 1 - 3 + 81 \cdot - 1 + 85$

Subtract $3$ from $- 1$ .

$- 4 + 81 \cdot - 1 + 85$

Multiply $81$ by $- 1$ .

$- 4 - 81 + 85$

Subtract $81$ from $- 4$ .

$- 85 + 85$

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.

$\frac{x^{3} - 3 x^{2} + 81 x + 85}{x + 1}$

Divide $x^{3} - 3 x^{2} + 81 x + 85$ 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 $0$ .

$x$

$1$

$x^{3}$

$3 x^{2}$

$81 x$

$85$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					-	$4 x^{2}$	-	$4 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$
							+	$85 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$
							+	$85 x$	+	$85$

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

				$x^{2}$	-	$4 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$
							+	$85 x$	+	$85$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$
							+	$85 x$	+	$85$
							+	$85 x$	+	$85$

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

				$x^{2}$	-	$4 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$
							+	$85 x$	+	$85$
							-	$85 x$	-	$85$

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

				$x^{2}$	-	$4 x$	+	$85$
$x$	+	$1$		$x^{3}$	-	$3 x^{2}$	+	$81 x$	+	$85$
			-	$x^{3}$	-	$x^{2}$
					-	$4 x^{2}$	+	$81 x$
					+	$4 x^{2}$	+	$4 x$
							+	$85 x$	+	$85$
							-	$85 x$	-	$85$
										$0$

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

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

Write $x^{3} - 3 x^{2} + 81 x + 85$ as a set of factors.

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

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

$x + 1 = 0$

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

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 $x^{2} - 4 x + 85$ 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 $\frac{4 \pm 18 i}{2}$ .

$x = 2 \pm 9 i$

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

$x = 2 \pm 9 i$

Change the $\pm$ to $+$ .

$x = 2 + 9 i$

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

$x = 2 \pm 9 i$

Change the $\pm$ to $-$ .

$x = 2 - 9 i$

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

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

Step 8

The polynomial can be written as a set of linear factors.

$(x + 1) (x - 2 - 9 i) (x - 2 + 9 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 2 x^{3} + 78 x^{2} + 166 x + 85$ .

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

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-2x^3+78x^2+166x+85

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