Find the Roots (Zeros) x^4+17x^3+60x^2-59x+77=0

$x^{4} + 17 x^{3} + 60 x^{2} - 59 x + 77 = 0$

Step 1

Factor the left side of the equation.

Factor $x^{4} + 17 x^{3} + 60 x^{2} - 59 x + 77$ 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.

$p = \pm 1, \pm 77, \pm 7, \pm 11 q = \pm 1$

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

$\pm 1, \pm 77, \pm 7, \pm 11$

Substitute $- 7$ and simplify the expression. In this case, the expression is equal to $0$ so $- 7$ is a root of the polynomial.

Substitute $- 7$ into the polynomial.

${(- 7)}^{4} + 17 {(- 7)}^{3} + 60 {(- 7)}^{2} - 59 \cdot - 7 + 77$

Raise $- 7$ to the power of $4$ .

$2401 + 17 {(- 7)}^{3} + 60 {(- 7)}^{2} - 59 \cdot - 7 + 77$

Raise $- 7$ to the power of $3$ .

$2401 + 17 \cdot - 343 + 60 {(- 7)}^{2} - 59 \cdot - 7 + 77$

Multiply $17$ by $- 343$ .

$2401 - 5831 + 60 {(- 7)}^{2} - 59 \cdot - 7 + 77$

Subtract $5831$ from $2401$ .

$- 3430 + 60 {(- 7)}^{2} - 59 \cdot - 7 + 77$

Raise $- 7$ to the power of $2$ .

$- 3430 + 60 \cdot 49 - 59 \cdot - 7 + 77$

Multiply $60$ by $49$ .

$- 3430 + 2940 - 59 \cdot - 7 + 77$

Add $- 3430$ and $2940$ .

$- 490 - 59 \cdot - 7 + 77$

Multiply $- 59$ by $- 7$ .

$- 490 + 413 + 77$

Add $- 490$ and $413$ .

$- 77 + 77$

Add $- 77$ and $77$ .

$0$

Since $- 7$ is a known root, divide the polynomial by $x + 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 17 x^{3} + 60 x^{2} - 59 x + 77}{x + 7}$

Divide $x^{4} + 17 x^{3} + 60 x^{2} - 59 x + 77$ by $x + 7$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

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

$x^{3}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 7 x^{3}$

$x^{3}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

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

$x^{3}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

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

$x^{3}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$10 x^{2}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$10 x^{2}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} + 70 x^{2}$

$x^{3}$

$10 x^{2}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

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

$x^{3}$

$10 x^{2}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

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

$x^{3}$

$10 x^{2}$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 70 x$

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

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

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

$11 x$

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

$x^{3}$

$10 x^{2}$

$10 x$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

$11 x$

$77$

Divide the highest order term in the dividend $11 x$ by the highest order term in divisor $x$ .

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

$11 x$

$77$

Multiply the new quotient term by the divisor.

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

$11 x$

$77$

$11 x$

$77$

The expression needs to be subtracted from the dividend, so change all the signs in $11 x + 77$

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

$11 x$

$77$

$11 x$

$77$

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

$x^{3}$

$10 x^{2}$

$10 x$

$11$

$x$

$7$

$x^{4}$

$17 x^{3}$

$60 x^{2}$

$59 x$

$77$

$x^{4}$

$7 x^{3}$

$10 x^{3}$

$60 x^{2}$

$10 x^{3}$

$70 x^{2}$

$10 x^{2}$

$59 x$

$10 x^{2}$

$70 x$

$11 x$

$77$

$11 x$

$77$

$0$

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

$x^{3} + 10 x^{2} - 10 x + 11$

Write $x^{4} + 17 x^{3} + 60 x^{2} - 59 x + 77$ as a set of factors.

$(x + 7) (x^{3} + 10 x^{2} - 10 x + 11) = 0$

Factor $x^{3} + 10 x^{2} - 10 x + 11$ 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.

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

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

$\pm 1, \pm 11$

Substitute $- 11$ and simplify the expression. In this case, the expression is equal to $0$ so $- 11$ is a root of the polynomial.

Substitute $- 11$ into the polynomial.

${(- 11)}^{3} + 10 {(- 11)}^{2} - 10 \cdot - 11 + 11$

Raise $- 11$ to the power of $3$ .

$- 1331 + 10 {(- 11)}^{2} - 10 \cdot - 11 + 11$

Raise $- 11$ to the power of $2$ .

$- 1331 + 10 \cdot 121 - 10 \cdot - 11 + 11$

Multiply $10$ by $121$ .

$- 1331 + 1210 - 10 \cdot - 11 + 11$

Add $- 1331$ and $1210$ .

$- 121 - 10 \cdot - 11 + 11$

Multiply $- 10$ by $- 11$ .

$- 121 + 110 + 11$

Add $- 121$ and $110$ .

$- 11 + 11$

Add $- 11$ and $11$ .

$0$

Since $- 11$ is a known root, divide the polynomial by $x + 11$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 10 x^{2} - 10 x + 11}{x + 11}$

Divide $x^{3} + 10 x^{2} - 10 x + 11$ by $x + 11$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$11$

$x^{3}$

$10 x^{2}$

$10 x$

$11$

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

					$x^{2}$
	$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			+	$x^{3}$	+	$11 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 11 x^{2}$

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$

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

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$

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

				$x^{2}$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$

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

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$11 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} - 11 x$

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$

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

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$

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

				$x^{2}$	-	$x$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$

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

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$
							+	$x$	+	$11$

The expression needs to be subtracted from the dividend, so change all the signs in $x + 11$

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$
							-	$x$	-	$11$

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

				$x^{2}$	-	$x$	+	$1$
$x$	+	$11$		$x^{3}$	+	$10 x^{2}$	-	$10 x$	+	$11$
			-	$x^{3}$	-	$11 x^{2}$
					-	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$11 x$
							+	$x$	+	$11$
							-	$x$	-	$11$
										$0$

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

$x^{2} - x + 1$

Write $x^{3} + 10 x^{2} - 10 x + 11$ as a set of factors.

$(x + 7) ((x + 11) (x^{2} - x + 1)) = 0$

Remove unnecessary parentheses.

$(x + 7) (x + 11) (x^{2} - x + 1) = 0$

Step 2

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

$x + 7 = 0$

$x + 11 = 0$

$x^{2} - x + 1 = 0$

Step 3

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 4

Set $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 5

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

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

$x^{2} - x + 1 = 0$

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $1$ .

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

Subtract $4$ from $1$ .

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

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

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

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

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

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

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 6

The final solution is all the values that make $(x + 7) (x + 11) (x^{2} - x + 1) = 0$ true.

$x = - 7, - 11, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 7

Find the Roots (Zeros) x^4+17x^3+60x^2-59x+77=0

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