Find the Roots/Zeros Using the Rational Roots Test 5x^5-44x^4+116x^3-110x^2+23x+10

$5 x^{5} - 44 x^{4} + 116 x^{3} - 110 x^{2} + 23 x + 10$

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.

$p = \pm 1, \pm 2, \pm 5, \pm 10$

$q = \pm 1, \pm 5$

Step 2

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

$\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}, \pm 5, \pm 10$

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.

$5 {(1)}^{5} - 44 {(1)}^{4} + 116 {(1)}^{3} - 110 {(1)}^{2} + 23 (1) + 10$

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.

One to any power is one.

$5 \cdot 1 - 44 {(1)}^{4} + 116 {(1)}^{3} - 110 {(1)}^{2} + 23 (1) + 10$

Multiply $5$ by $1$ .

$5 - 44 {(1)}^{4} + 116 {(1)}^{3} - 110 {(1)}^{2} + 23 (1) + 10$

One to any power is one.

$5 - 44 \cdot 1 + 116 {(1)}^{3} - 110 {(1)}^{2} + 23 (1) + 10$

Multiply $- 44$ by $1$ .

$5 - 44 + 116 {(1)}^{3} - 110 {(1)}^{2} + 23 (1) + 10$

One to any power is one.

$5 - 44 + 116 \cdot 1 - 110 {(1)}^{2} + 23 (1) + 10$

Multiply $116$ by $1$ .

$5 - 44 + 116 - 110 {(1)}^{2} + 23 (1) + 10$

One to any power is one.

$5 - 44 + 116 - 110 \cdot 1 + 23 (1) + 10$

Multiply $- 110$ by $1$ .

$5 - 44 + 116 - 110 + 23 (1) + 10$

Multiply $23$ by $1$ .

$5 - 44 + 116 - 110 + 23 + 10$

Simplify by adding and subtracting.

Subtract $44$ from $5$ .

$- 39 + 116 - 110 + 23 + 10$

Add $- 39$ and $116$ .

$77 - 110 + 23 + 10$

Subtract $110$ from $77$ .

$- 33 + 23 + 10$

Add $- 33$ and $23$ .

$- 10 + 10$

Add $- 10$ and $10$ .

$0$

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.

$\frac{5 x^{5} - 44 x^{4} + 116 x^{3} - 110 x^{2} + 23 x + 10}{x - 1}$

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.

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0					\xa0

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0					\xa0
	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(1)$ and place the result of $(5)$ under the next term in the dividend $(- 44)$ .

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$				\xa0
	$5$

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

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$				\xa0
	$5$	$- 39$

Multiply the newest entry in the result $(- 39)$ by the divisor $(1)$ and place the result of $(- 39)$ under the next term in the dividend $(116)$ .

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$			\xa0
	$5$	$- 39$

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

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$			\xa0
	$5$	$- 39$	$77$

Multiply the newest entry in the result $(77)$ by the divisor $(1)$ and place the result of $(77)$ under the next term in the dividend $(- 110)$ .

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$	$77$		\xa0
	$5$	$- 39$	$77$

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

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$	$77$		\xa0
	$5$	$- 39$	$77$	$- 33$

Multiply the newest entry in the result $(- 33)$ by the divisor $(1)$ and place the result of $(- 33)$ under the next term in the dividend $(23)$ .

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$	$77$	$- 33$	\xa0
	$5$	$- 39$	$77$	$- 33$

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

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$	$77$	$- 33$	\xa0
	$5$	$- 39$	$77$	$- 33$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(1)$ and place the result of $(- 10)$ under the next term in the dividend $(10)$ .

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$	$77$	$- 33$	$- 10$
	$5$	$- 39$	$77$	$- 33$	$- 10$

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

$1$	$5$	$- 44$	$116$	$- 110$	$23$	$10$
	\xa0	$5$	$- 39$	$77$	$- 33$	$- 10$
	$5$	$- 39$	$77$	$- 33$	$- 10$	$0$

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

$5 x^{4} + - 39 x^{3} + 77 x^{2} + - 33 x - 10$

Simplify the quotient polynomial.

$5 x^{4} - 39 x^{3} + 77 x^{2} - 33 x - 10$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $5 x^{4} - 39 x^{3} + 77 x^{2} - 33 x - 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1, \pm 5$

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

$\pm 1, \pm 0.2, \pm 10, \pm 2, \pm 0.4, \pm 5$

Substitute $- 0.2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.2$ is a root of the polynomial.

Substitute $- 0.2$ into the polynomial.

$5 {(- 0.2)}^{4} - 39 {(- 0.2)}^{3} + 77 {(- 0.2)}^{2} - 33 \cdot - 0.2 - 10$

Raise $- 0.2$ to the power of $4$ .

$5 \cdot 0.0016 - 39 {(- 0.2)}^{3} + 77 {(- 0.2)}^{2} - 33 \cdot - 0.2 - 10$

Multiply $5$ by $0.0016$ .

$0.008 - 39 {(- 0.2)}^{3} + 77 {(- 0.2)}^{2} - 33 \cdot - 0.2 - 10$

Raise $- 0.2$ to the power of $3$ .

$0.008 - 39 \cdot - 0.008 + 77 {(- 0.2)}^{2} - 33 \cdot - 0.2 - 10$

Multiply $- 39$ by $- 0.008$ .

$0.008 + 0.312 + 77 {(- 0.2)}^{2} - 33 \cdot - 0.2 - 10$

Add $0.008$ and $0.312$ .

$0.32 + 77 {(- 0.2)}^{2} - 33 \cdot - 0.2 - 10$

Raise $- 0.2$ to the power of $2$ .

$0.32 + 77 \cdot 0.04 - 33 \cdot - 0.2 - 10$

Multiply $77$ by $0.04$ .

$0.32 + 3.08 - 33 \cdot - 0.2 - 10$

Add $0.32$ and $3.08$ .

$3.4 - 33 \cdot - 0.2 - 10$

Multiply $- 33$ by $- 0.2$ .

$3.4 + 6.6 - 10$

Add $3.4$ and $6.6$ .

$10 - 10$

Subtract $10$ from $10$ .

$0$

Since $- 0.2$ is a known root, divide the polynomial by $5 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{4} - 39 x^{3} + 77 x^{2} - 33 x - 10}{5 x + 1}$

Divide $5 x^{4} - 39 x^{3} + 77 x^{2} - 33 x - 10$ by $5 x + 1$ .

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

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $5 x$ .

$x^{3}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + x^{3}$

				$x^{3}$
$5 x$	+	$1$		$5 x^{4}$	-	$39 x^{3}$	+	$77 x^{2}$	-	$33 x$	-	$10$
			-	$5 x^{4}$	-	$x^{3}$

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

				$x^{3}$
$5 x$	+	$1$		$5 x^{4}$	-	$39 x^{3}$	+	$77 x^{2}$	-	$33 x$	-	$10$
			-	$5 x^{4}$	-	$x^{3}$
					-	$40 x^{3}$

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

$x^{3}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

Divide the highest order term in the dividend $- 40 x^{3}$ by the highest order term in divisor $5 x$ .

$x^{3}$

$8 x^{2}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$8 x^{2}$
$5 x$	+	$1$		$5 x^{4}$	-	$39 x^{3}$	+	$77 x^{2}$	-	$33 x$	-	$10$
			-	$5 x^{4}$	-	$x^{3}$
					-	$40 x^{3}$	+	$77 x^{2}$
					-	$40 x^{3}$	-	$8 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x^{3} - 8 x^{2}$

$x^{3}$

$8 x^{2}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

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

$x^{3}$

$8 x^{2}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

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

$x^{3}$

$8 x^{2}$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

Divide the highest order term in the dividend $85 x^{2}$ by the highest order term in divisor $5 x$ .

$x^{3}$

$8 x^{2}$

$17 x$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$8 x^{2}$

$17 x$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

The expression needs to be subtracted from the dividend, so change all the signs in $85 x^{2} + 17 x$

$x^{3}$

$8 x^{2}$

$17 x$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

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

$x^{3}$

$8 x^{2}$

$17 x$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

$50 x$

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

$x^{3}$

$8 x^{2}$

$17 x$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

$50 x$

$10$

Divide the highest order term in the dividend $- 50 x$ by the highest order term in divisor $5 x$ .

$x^{3}$

$8 x^{2}$

$17 x$

$10$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

$50 x$

$10$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
$5 x$	+	$1$		$5 x^{4}$	-	$39 x^{3}$	+	$77 x^{2}$	-	$33 x$	-	$10$
			-	$5 x^{4}$	-	$x^{3}$
					-	$40 x^{3}$	+	$77 x^{2}$
					+	$40 x^{3}$	+	$8 x^{2}$
							+	$85 x^{2}$	-	$33 x$
							-	$85 x^{2}$	-	$17 x$
									-	$50 x$	-	$10$
									-	$50 x$	-	$10$

The expression needs to be subtracted from the dividend, so change all the signs in $- 50 x - 10$

$x^{3}$

$8 x^{2}$

$17 x$

$10$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

$50 x$

$10$

$50 x$

$10$

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

$x^{3}$

$8 x^{2}$

$17 x$

$10$

$5 x$

$1$

$5 x^{4}$

$39 x^{3}$

$77 x^{2}$

$33 x$

$10$

$5 x^{4}$

$x^{3}$

$40 x^{3}$

$77 x^{2}$

$40 x^{3}$

$8 x^{2}$

$85 x^{2}$

$33 x$

$85 x^{2}$

$17 x$

$50 x$

$10$

$50 x$

$10$

$0$

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

$x^{3} - 8 x^{2} + 17 x - 10$

Write $5 x^{4} - 39 x^{3} + 77 x^{2} - 33 x - 10$ as a set of factors.

$(5 x + 1) (x^{3} - 8 x^{2} + 17 x - 10) = 0$

Factor $x^{3} - 8 x^{2} + 17 x - 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1$

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

$\pm 1, \pm 10, \pm 2, \pm 5$

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.

$1^{3} - 8 \cdot 1^{2} + 17 \cdot 1 - 10$

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

$1 - 8 \cdot 1^{2} + 17 \cdot 1 - 10$

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

$1 - 8 \cdot 1 + 17 \cdot 1 - 10$

Multiply $- 8$ by $1$ .

$1 - 8 + 17 \cdot 1 - 10$

Subtract $8$ from $1$ .

$- 7 + 17 \cdot 1 - 10$

Multiply $17$ by $1$ .

$- 7 + 17 - 10$

Add $- 7$ and $17$ .

$10 - 10$

Subtract $10$ from $10$ .

$0$

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.

$\frac{x^{3} - 8 x^{2} + 17 x - 10}{x - 1}$

Divide $x^{3} - 8 x^{2} + 17 x - 10$ 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 $0$ .

$x$

$1$

$x^{3}$

$8 x^{2}$

$17 x$

$10$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					-	$7 x^{2}$	+	$7 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} + 7 x$

				$x^{2}$	-	$7 x$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$

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

				$x^{2}$	-	$7 x$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$
							+	$10 x$

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

				$x^{2}$	-	$7 x$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$
							+	$10 x$	-	$10$

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$
							+	$10 x$	-	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$
							+	$10 x$	-	$10$
							+	$10 x$	-	$10$

The expression needs to be subtracted from the dividend, so change all the signs in $10 x - 10$

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$
							+	$10 x$	-	$10$
							-	$10 x$	+	$10$

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

				$x^{2}$	-	$7 x$	+	$10$
$x$	-	$1$		$x^{3}$	-	$8 x^{2}$	+	$17 x$	-	$10$
			-	$x^{3}$	+	$x^{2}$
					-	$7 x^{2}$	+	$17 x$
					+	$7 x^{2}$	-	$7 x$
							+	$10 x$	-	$10$
							-	$10 x$	+	$10$
										$0$

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

$x^{2} - 7 x + 10$

Write $x^{3} - 8 x^{2} + 17 x - 10$ as a set of factors.

$(5 x + 1) ((x - 1) (x^{2} - 7 x + 10)) = 0$

Factor $x^{2} - 7 x + 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$(5 x + 1) ((x - 1) ((x - 5) (x - 2))) = 0$

Remove unnecessary parentheses.

$(5 x + 1) ((x - 1) (x - 5) (x - 2)) = 0$

Remove unnecessary parentheses.

$(5 x + 1) (x - 1) (x - 5) (x - 2) = 0$

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

$5 x + 1 = 0$

$x - 1 = 0$

$x - 5 = 0$

$x - 2 = 0$

Set $5 x + 1$ equal to $0$ and solve for $x$ .

Set $5 x + 1$ equal to $0$ .

$5 x + 1 = 0$

Solve $5 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Divide each term in $5 x = - 1$ by $5$ and simplify.

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(5 x + 1) (x - 1) (x - 5) (x - 2) = 0$ true.

$x = - \frac{1}{5}, 1, 5, 2$

Step 8

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

$(x - 1) (x + \frac{1}{5}) (x - 5) (x - 2)$

Step 9

These are the roots (zeros) of the polynomial $5 x^{5} - 44 x^{4} + 116 x^{3} - 110 x^{2} + 23 x + 10$ .

$x = 1, - \frac{1}{5}, 5, 2$

Step 10

Find the Roots/Zeros Using the Rational Roots Test 5x^5-44x^4+116x^3-110x^2+23x+10

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