Find the Roots/Zeros Using the Rational Roots Test f(x)=5x^3-18x^2-11x+12

$f (x) = 5 x^{3} - 18 x^{2} - 11 x + 12$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

$q = \pm 1, \pm 5$

Step 2

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

$\pm 1, \pm \frac{1}{5}, \pm 2, \pm \frac{2}{5}, \pm 3, \pm \frac{3}{5}, \pm 4, \pm \frac{4}{5}, \pm 6, \pm \frac{6}{5}, \pm 12, \pm \frac{12}{5}$

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.

$5 {(- 1)}^{3} - 18 {(- 1)}^{2} - 11 \cdot - 1 + 12$

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 $3$ .

$5 \cdot - 1 - 18 {(- 1)}^{2} - 11 \cdot - 1 + 12$

Multiply $5$ by $- 1$ .

$- 5 - 18 {(- 1)}^{2} - 11 \cdot - 1 + 12$

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

$- 5 - 18 \cdot 1 - 11 \cdot - 1 + 12$

Multiply $- 18$ by $1$ .

$- 5 - 18 - 11 \cdot - 1 + 12$

Multiply $- 11$ by $- 1$ .

$- 5 - 18 + 11 + 12$

Simplify by adding and subtracting.

Subtract $18$ from $- 5$ .

$- 23 + 11 + 12$

Add $- 23$ and $11$ .

$- 12 + 12$

Add $- 12$ and $12$ .

$0$

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.

$\frac{5 x^{3} - 18 x^{2} - 11 x + 12}{x + 1}$

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.

$- 1$	$5$	$- 18$	$- 11$	$12$
	\xa0			\xa0

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

$- 1$	$5$	$- 18$	$- 11$	$12$
	\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 $(- 18)$ .

$- 1$	$5$	$- 18$	$- 11$	$12$
	\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.

$- 1$	$5$	$- 18$	$- 11$	$12$
	\xa0	$- 5$		\xa0
	$5$	$- 23$

Multiply the newest entry in the result $(- 23)$ by the divisor $(- 1)$ and place the result of $(23)$ under the next term in the dividend $(- 11)$ .

$- 1$	$5$	$- 18$	$- 11$	$12$
	\xa0	$- 5$	$23$	\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.

$- 1$	$5$	$- 18$	$- 11$	$12$
	\xa0	$- 5$	$23$	\xa0
	$5$	$- 23$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(- 1)$ and place the result of $(- 12)$ under the next term in the dividend $(12)$ .

$- 1$	$5$	$- 18$	$- 11$	$12$
	\xa0	$- 5$	$23$	$- 12$
	$5$	$- 23$	$12$

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

$- 1$	$5$	$- 18$	$- 11$	$12$
	\xa0	$- 5$	$23$	$- 12$
	$5$	$- 23$	$12$	$0$

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

$5 x^{2} + - 23 x + 12$

Simplify the quotient polynomial.

$5 x^{2} - 23 x + 12$

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 $a \cdot c = 5 \cdot 12 = 60$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 x$ .

$(x + 1) + 5 x^{2} - 23 x + 12$

Rewrite $- 23$ as $- 3$ plus $- 20$

$(x + 1) + 5 x^{2} + (- 3 - 20) x + 12$

Apply the distributive property.

$(x + 1) + 5 x^{2} - 3 x - 20 x + 12$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) (5 x^{2} - 3 x) - 20 x + 12$

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

$(x + 1) + x (5 x - 3) - 4 (5 x - 3)$

Factor the polynomial by factoring out the greatest common factor, $5 x - 3$ .

$(x + 1) + (5 x - 3) (x - 4)$

$(x + 1) (5 x - 3) (x - 4)$

Step 8

Factor the left side of the equation.

Factor $5 x^{3} - 18 x^{2} - 11 x + 12$ 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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4 q = \pm 1, \pm 5$

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

$\pm 1, \pm 0.2, \pm 12, \pm 2.4, \pm 2, \pm 0.4, \pm 6, \pm 1.2, \pm 3, \pm 0.6, \pm 4, \pm 0.8$

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.

$5 {(- 1)}^{3} - 18 {(- 1)}^{2} - 11 \cdot - 1 + 12$

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

$5 \cdot - 1 - 18 {(- 1)}^{2} - 11 \cdot - 1 + 12$

Multiply $5$ by $- 1$ .

$- 5 - 18 {(- 1)}^{2} - 11 \cdot - 1 + 12$

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

$- 5 - 18 \cdot 1 - 11 \cdot - 1 + 12$

Multiply $- 18$ by $1$ .

$- 5 - 18 - 11 \cdot - 1 + 12$

Subtract $18$ from $- 5$ .

$- 23 - 11 \cdot - 1 + 12$

Multiply $- 11$ by $- 1$ .

$- 23 + 11 + 12$

Add $- 23$ and $11$ .

$- 12 + 12$

Add $- 12$ and $12$ .

$0$

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.

$\frac{5 x^{3} - 18 x^{2} - 11 x + 12}{x + 1}$

Divide $5 x^{3} - 18 x^{2} - 11 x + 12$ 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 $0$ .

$x$

$1$

$5 x^{3}$

$18 x^{2}$

$11 x$

$12$

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

					$5 x^{2}$
	$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			+	$5 x^{3}$	+	$5 x^{2}$

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

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$

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

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$

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

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$

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

				$5 x^{2}$	-	$23 x$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$23 x$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					-	$23 x^{2}$	-	$23 x$

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

				$5 x^{2}$	-	$23 x$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$

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

				$5 x^{2}$	-	$23 x$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$
							+	$12 x$

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

				$5 x^{2}$	-	$23 x$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$
							+	$12 x$	+	$12$

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $x$ .

				$5 x^{2}$	-	$23 x$	+	$12$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$
							+	$12 x$	+	$12$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	-	$23 x$	+	$12$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$
							+	$12 x$	+	$12$
							+	$12 x$	+	$12$

The expression needs to be subtracted from the dividend, so change all the signs in $12 x + 12$

				$5 x^{2}$	-	$23 x$	+	$12$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$
							+	$12 x$	+	$12$
							-	$12 x$	-	$12$

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

				$5 x^{2}$	-	$23 x$	+	$12$
$x$	+	$1$		$5 x^{3}$	-	$18 x^{2}$	-	$11 x$	+	$12$
			-	$5 x^{3}$	-	$5 x^{2}$
					-	$23 x^{2}$	-	$11 x$
					+	$23 x^{2}$	+	$23 x$
							+	$12 x$	+	$12$
							-	$12 x$	-	$12$
										$0$

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

$5 x^{2} - 23 x + 12$

Write $5 x^{3} - 18 x^{2} - 11 x + 12$ as a set of factors.

$(x + 1) (5 x^{2} - 23 x + 12) = 0$

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 $a \cdot c = 5 \cdot 12 = 60$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 x$ .

$(x + 1) (5 x^{2} - 23 x + 12) = 0$

Rewrite $- 23$ as $- 3$ plus $- 20$

$(x + 1) (5 x^{2} + (- 3 - 20) x + 12) = 0$

Apply the distributive property.

$(x + 1) (5 x^{2} - 3 x - 20 x + 12) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) ((5 x^{2} - 3 x) - 20 x + 12) = 0$

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

$(x + 1) (x (5 x - 3) - 4 (5 x - 3)) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x - 3$ .

$(x + 1) ((5 x - 3) (x - 4)) = 0$

Remove unnecessary parentheses.

$(x + 1) (5 x - 3) (x - 4) = 0$

Step 9

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

$x + 1 = 0$

$5 x - 3 = 0$

$x - 4 = 0$

Step 10

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.

$x = - 1$

Step 11

Set $5 x - 3$ equal to $0$ and solve for $x$ .

Set $5 x - 3$ equal to $0$ .

$5 x - 3 = 0$

Solve $5 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$5 x = 3$

Divide each term in $5 x = 3$ by $5$ and simplify.

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 12

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 13

The final solution is all the values that make $(x + 1) (5 x - 3) (x - 4) = 0$ true.

$x = - 1, \frac{3}{5}, 4$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=5x^3-18x^2-11x+12

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