Find the Roots/Zeros Using the Rational Roots Test 5x^3+13x^2-56x+20

$5 x^{3} + 13 x^{2} - 56 x + 20$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$

$q = \pm 1, \pm 5$

Step 2

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

$\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}, \pm 4, \pm \frac{4}{5}, \pm 5, \pm 10, \pm 20$

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.

$5 {(2)}^{3} + 13 {(2)}^{2} - 56 \cdot 2 + 20$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ is a root of the polynomial.

Simplify each term.

Raise $2$ to the power of $3$ .

$5 \cdot 8 + 13 {(2)}^{2} - 56 \cdot 2 + 20$

Multiply $5$ by $8$ .

$40 + 13 {(2)}^{2} - 56 \cdot 2 + 20$

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

$40 + 13 \cdot 4 - 56 \cdot 2 + 20$

Multiply $13$ by $4$ .

$40 + 52 - 56 \cdot 2 + 20$

Multiply $- 56$ by $2$ .

$40 + 52 - 112 + 20$

Simplify by adding and subtracting.

Add $40$ and $52$ .

$92 - 112 + 20$

Subtract $112$ from $92$ .

$- 20 + 20$

Add $- 20$ and $20$ .

$0$

Step 5

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{3} + 13 x^{2} - 56 x + 20}{x - 2}$

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.

$2$	$5$	$13$	$- 56$	$20$
	\xa0			\xa0

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

$2$	$5$	$13$	$- 56$	$20$
	\xa0			\xa0
	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(2)$ and place the result of $(10)$ under the next term in the dividend $(13)$ .

$2$	$5$	$13$	$- 56$	$20$
	\xa0	$10$		\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.

$2$	$5$	$13$	$- 56$	$20$
	\xa0	$10$		\xa0
	$5$	$23$

Multiply the newest entry in the result $(23)$ by the divisor $(2)$ and place the result of $(46)$ under the next term in the dividend $(- 56)$ .

$2$	$5$	$13$	$- 56$	$20$
	\xa0	$10$	$46$	\xa0
	$5$	$23$

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

$2$	$5$	$13$	$- 56$	$20$
	\xa0	$10$	$46$	\xa0
	$5$	$23$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(2)$ and place the result of $(- 20)$ under the next term in the dividend $(20)$ .

$2$	$5$	$13$	$- 56$	$20$
	\xa0	$10$	$46$	$- 20$
	$5$	$23$	$- 10$

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

$2$	$5$	$13$	$- 56$	$20$
	\xa0	$10$	$46$	$- 20$
	$5$	$23$	$- 10$	$0$

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

$5 x^{2} + 23 x - 10$

Step 7

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 10 = - 50$ and whose sum is $b = 23$ .

Factor $23$ out of $23 x$ .

$(x - 2) + 5 x^{2} + 23 (x) - 10$

Rewrite $23$ as $- 2$ plus $25$

$(x - 2) + 5 x^{2} + (- 2 + 25) x - 10$

Apply the distributive property.

$(x - 2) + 5 x^{2} - 2 x + 25 x - 10$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 2) (5 x^{2} - 2 x) + 25 x - 10$

Factor out the greatest common factor (GCF) from each group.

$(x - 2) + x (5 x - 2) + 5 (5 x - 2)$

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

$(x - 2) + (5 x - 2) (x + 5)$

$(x - 2) (5 x - 2) (x + 5)$

Step 8

Factor the left side of the equation.

Factor $5 x^{3} + 13 x^{2} - 56 x + 20$ 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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5 q = \pm 1, \pm 5$

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

$\pm 1, \pm 0.2, \pm 20, \pm 4, \pm 2, \pm 0.4, \pm 10, \pm 0.8, \pm 5$

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$5 \cdot 2^{3} + 13 \cdot 2^{2} - 56 \cdot 2 + 20$

Raise $2$ to the power of $3$ .

$5 \cdot 8 + 13 \cdot 2^{2} - 56 \cdot 2 + 20$

Multiply $5$ by $8$ .

$40 + 13 \cdot 2^{2} - 56 \cdot 2 + 20$

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

$40 + 13 \cdot 4 - 56 \cdot 2 + 20$

Multiply $13$ by $4$ .

$40 + 52 - 56 \cdot 2 + 20$

Add $40$ and $52$ .

$92 - 56 \cdot 2 + 20$

Multiply $- 56$ by $2$ .

$92 - 112 + 20$

Subtract $112$ from $92$ .

$- 20 + 20$

Add $- 20$ and $20$ .

$0$

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{3} + 13 x^{2} - 56 x + 20}{x - 2}$

Divide $5 x^{3} + 13 x^{2} - 56 x + 20$ by $x - 2$ .

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

$x$

$2$

$5 x^{3}$

$13 x^{2}$

$56 x$

$20$

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

					$5 x^{2}$
	$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			+	$5 x^{3}$	-	$10 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 10 x^{2}$

				$5 x^{2}$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$

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

				$5 x^{2}$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$

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

				$5 x^{2}$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$

Divide the highest order term in the dividend $23 x^{2}$ by the highest order term in divisor $x$ .

				$5 x^{2}$	+	$23 x$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$23 x$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					+	$23 x^{2}$	-	$46 x$

The expression needs to be subtracted from the dividend, so change all the signs in $23 x^{2} - 46 x$

				$5 x^{2}$	+	$23 x$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$

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

				$5 x^{2}$	+	$23 x$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$
							-	$10 x$

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

				$5 x^{2}$	+	$23 x$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$
							-	$10 x$	+	$20$

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

				$5 x^{2}$	+	$23 x$	-	$10$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$
							-	$10 x$	+	$20$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$23 x$	-	$10$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$
							-	$10 x$	+	$20$
							-	$10 x$	+	$20$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x + 20$

				$5 x^{2}$	+	$23 x$	-	$10$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$
							-	$10 x$	+	$20$
							+	$10 x$	-	$20$

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

				$5 x^{2}$	+	$23 x$	-	$10$
$x$	-	$2$		$5 x^{3}$	+	$13 x^{2}$	-	$56 x$	+	$20$
			-	$5 x^{3}$	+	$10 x^{2}$
					+	$23 x^{2}$	-	$56 x$
					-	$23 x^{2}$	+	$46 x$
							-	$10 x$	+	$20$
							+	$10 x$	-	$20$
										$0$

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

$5 x^{2} + 23 x - 10$

Write $5 x^{3} + 13 x^{2} - 56 x + 20$ as a set of factors.

$(x - 2) (5 x^{2} + 23 x - 10) = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 10 = - 50$ and whose sum is $b = 23$ .

Factor $23$ out of $23 x$ .

$(x - 2) (5 x^{2} + 23 (x) - 10) = 0$

Rewrite $23$ as $- 2$ plus $25$

$(x - 2) (5 x^{2} + (- 2 + 25) x - 10) = 0$

Apply the distributive property.

$(x - 2) (5 x^{2} - 2 x + 25 x - 10) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 2) ((5 x^{2} - 2 x) + 25 x - 10) = 0$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

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

Remove unnecessary parentheses.

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

Step 9

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

$x - 2 = 0$

$5 x - 2 = 0$

$x + 5 = 0$

Step 10

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.

$x = 2$

Step 11

Set $5 x - 2$ equal to $0$ and solve for $x$ .

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Solve $5 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$5 x = 2$

Divide each term in $5 x = 2$ by $5$ and simplify.

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 12

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

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 13

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

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

Step 14

Find the Roots/Zeros Using the Rational Roots Test 5x^3+13x^2-56x+20

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