Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-6x^3+50x^2+122x+65

$f (x) = x^{4} - 6 x^{3} + 50 x^{2} + 122 x + 65$

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.

$p = \pm 1, \pm 5, \pm 13, \pm 65$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 13, \pm 65$

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.

${(- 1)}^{4} - 6 {(- 1)}^{3} + 50 {(- 1)}^{2} + 122 (- 1) + 65$

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 $4$ .

$1 - 6 {(- 1)}^{3} + 50 {(- 1)}^{2} + 122 (- 1) + 65$

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

$1 - 6 \cdot - 1 + 50 {(- 1)}^{2} + 122 (- 1) + 65$

Multiply $- 6$ by $- 1$ .

$1 + 6 + 50 {(- 1)}^{2} + 122 (- 1) + 65$

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

$1 + 6 + 50 \cdot 1 + 122 (- 1) + 65$

Multiply $50$ by $1$ .

$1 + 6 + 50 + 122 (- 1) + 65$

Multiply $122$ by $- 1$ .

$1 + 6 + 50 - 122 + 65$

Simplify by adding and subtracting.

Add $1$ and $6$ .

$7 + 50 - 122 + 65$

Add $7$ and $50$ .

$57 - 122 + 65$

Subtract $122$ from $57$ .

$- 65 + 65$

Add $- 65$ and $65$ .

$0$

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.

$\frac{x^{4} - 6 x^{3} + 50 x^{2} + 122 x + 65}{x + 1}$

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.

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0				\xa0

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

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\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 $(- 6)$ .

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\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.

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$			\xa0
	$1$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 1)$ and place the result of $(7)$ under the next term in the dividend $(50)$ .

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$	$7$		\xa0
	$1$	$- 7$

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

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$	$7$		\xa0
	$1$	$- 7$	$57$

Multiply the newest entry in the result $(57)$ by the divisor $(- 1)$ and place the result of $(- 57)$ under the next term in the dividend $(122)$ .

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$	$7$	$- 57$	\xa0
	$1$	$- 7$	$57$

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

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$	$7$	$- 57$	\xa0
	$1$	$- 7$	$57$	$65$

Multiply the newest entry in the result $(65)$ by the divisor $(- 1)$ and place the result of $(- 65)$ under the next term in the dividend $(65)$ .

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$	$7$	$- 57$	$- 65$
	$1$	$- 7$	$57$	$65$

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

$- 1$	$1$	$- 6$	$50$	$122$	$65$
	\xa0	$- 1$	$7$	$- 57$	$- 65$
	$1$	$- 7$	$57$	$65$	$0$

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

$1 x^{3} + - 7 x^{2} + (57) x + 65$

Simplify the quotient polynomial.

$x^{3} - 7 x^{2} + 57 x + 65$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 7 x^{2} + 57 x + 65$ 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.

$p = \pm 1, \pm 65, \pm 5, \pm 13 q = \pm 1$

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

$\pm 1, \pm 65, \pm 5, \pm 13$

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.

${(- 1)}^{3} - 7 {(- 1)}^{2} + 57 \cdot - 1 + 65$

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

$- 1 - 7 {(- 1)}^{2} + 57 \cdot - 1 + 65$

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

$- 1 - 7 \cdot 1 + 57 \cdot - 1 + 65$

Multiply $- 7$ by $1$ .

$- 1 - 7 + 57 \cdot - 1 + 65$

Subtract $7$ from $- 1$ .

$- 8 + 57 \cdot - 1 + 65$

Multiply $57$ by $- 1$ .

$- 8 - 57 + 65$

Subtract $57$ from $- 8$ .

$- 65 + 65$

Add $- 65$ and $65$ .

$0$

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.

$\frac{x^{3} - 7 x^{2} + 57 x + 65}{x + 1}$

Divide $x^{3} - 7 x^{2} + 57 x + 65$ 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 $0$ .

$x$

$1$

$x^{3}$

$7 x^{2}$

$57 x$

$65$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					-	$8 x^{2}$	-	$8 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$
							+	$65 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$
							+	$65 x$	+	$65$

Divide the highest order term in the dividend $65 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$
							+	$65 x$	+	$65$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$
							+	$65 x$	+	$65$
							+	$65 x$	+	$65$

The expression needs to be subtracted from the dividend, so change all the signs in $65 x + 65$

				$x^{2}$	-	$8 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$
							+	$65 x$	+	$65$
							-	$65 x$	-	$65$

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

				$x^{2}$	-	$8 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$7 x^{2}$	+	$57 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$57 x$
					+	$8 x^{2}$	+	$8 x$
							+	$65 x$	+	$65$
							-	$65 x$	-	$65$
										$0$

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

$x^{2} - 8 x + 65$

Write $x^{3} - 7 x^{2} + 57 x + 65$ as a set of factors.

$(x + 1) (x^{2} - 8 x + 65) = 0$

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

$x + 1 = 0$

$x^{2} - 8 x + 65 = 0$

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 $x^{2} - 8 x + 65$ equal to $0$ and solve for $x$ .

Set $x^{2} - 8 x + 65$ equal to $0$ .

$x^{2} - 8 x + 65 = 0$

Solve $x^{2} - 8 x + 65 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 8$ , and $c = 65$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 65)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 65}}{2 \cdot 1}$

Multiply $- 4$ by $65$ .

$x = \frac{8 \pm \sqrt{64 - 260}}{2 \cdot 1}$

Subtract $260$ from $64$ .

$x = \frac{8 \pm \sqrt{- 196}}{2 \cdot 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot 14}{2 \cdot 1}$

Move $14$ to the left of $i$ .

$x = \frac{8 \pm 14 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 14 i}{2}$

Simplify $\frac{8 \pm 14 i}{2}$ .

$x = 4 \pm 7 i$

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 65}}{2 \cdot 1}$

Multiply $- 4$ by $65$ .

$x = \frac{8 \pm \sqrt{64 - 260}}{2 \cdot 1}$

Subtract $260$ from $64$ .

$x = \frac{8 \pm \sqrt{- 196}}{2 \cdot 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot 14}{2 \cdot 1}$

Move $14$ to the left of $i$ .

$x = \frac{8 \pm 14 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 14 i}{2}$

Simplify $\frac{8 \pm 14 i}{2}$ .

$x = 4 \pm 7 i$

Change the $\pm$ to $+$ .

$x = 4 + 7 i$

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 65}}{2 \cdot 1}$

Multiply $- 4$ by $65$ .

$x = \frac{8 \pm \sqrt{64 - 260}}{2 \cdot 1}$

Subtract $260$ from $64$ .

$x = \frac{8 \pm \sqrt{- 196}}{2 \cdot 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

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

$x = \frac{8 \pm i \cdot 14}{2 \cdot 1}$

Move $14$ to the left of $i$ .

$x = \frac{8 \pm 14 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 14 i}{2}$

Simplify $\frac{8 \pm 14 i}{2}$ .

$x = 4 \pm 7 i$

Change the $\pm$ to $-$ .

$x = 4 - 7 i$

The final answer is the combination of both solutions.

$x = 4 + 7 i, 4 - 7 i$

The final solution is all the values that make $(x + 1) (x^{2} - 8 x + 65) = 0$ true.

$x = - 1, 4 + 7 i, 4 - 7 i$

Step 8

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

$(x + 1) (x - 4 - 7 i) (x - 4 + 7 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 6 x^{3} + 50 x^{2} + 122 x + 65$ .

$x = - 1, 4 + 7 i, 4 - 7 i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-6x^3+50x^2+122x+65

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