Find the Roots/Zeros Using the Rational Roots Test f(x)=6x^3+31x^2+4x-5

$f (x) = 6 x^{3} + 31 x^{2} + 4 x - 5$

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.

$p = \pm 1, \pm 5$

$q = \pm 1, \pm 2, \pm 3, \pm 6$

Step 2

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

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

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.

$6 {(- \frac{1}{2})}^{3} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{1}{2}$ is a root of the polynomial.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{2}$ .

$6 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Apply the product rule to $\frac{1}{2}$ .

$6 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

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

$6 (- \frac{1^{3}}{2^{3}}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

One to any power is one.

$6 (- \frac{1}{2^{3}}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

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

$6 (- \frac{1}{8}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$6 (\frac{- 1}{8}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Factor $2$ out of $6$ .

$2 (3) \frac{- 1}{8} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Factor $2$ out of $8$ .

$2 \cdot 3 \frac{- 1}{2 \cdot 4} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Cancel the common factor.

$2 \cdot 3 \frac{- 1}{2 \cdot 4} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Rewrite the expression.

$3 (\frac{- 1}{4}) + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Combine $3$ and $\frac{- 1}{4}$ .

$\frac{3 \cdot - 1}{4} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Multiply $3$ by $- 1$ .

$\frac{- 3}{4} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Move the negative in front of the fraction.

$- \frac{3}{4} + 31 {(- \frac{1}{2})}^{2} + 4 (- \frac{1}{2}) - 5$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{2}$ .

$- \frac{3}{4} + 31 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + 4 (- \frac{1}{2}) - 5$

Apply the product rule to $\frac{1}{2}$ .

$- \frac{3}{4} + 31 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) + 4 (- \frac{1}{2}) - 5$

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

$- \frac{3}{4} + 31 (1 \frac{1^{2}}{2^{2}}) + 4 (- \frac{1}{2}) - 5$

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$- \frac{3}{4} + 31 \frac{1^{2}}{2^{2}} + 4 (- \frac{1}{2}) - 5$

One to any power is one.

$- \frac{3}{4} + 31 \frac{1}{2^{2}} + 4 (- \frac{1}{2}) - 5$

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

$- \frac{3}{4} + 31 (\frac{1}{4}) + 4 (- \frac{1}{2}) - 5$

Combine $31$ and $\frac{1}{4}$ .

$- \frac{3}{4} + \frac{31}{4} + 4 (- \frac{1}{2}) - 5$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{3}{4} + \frac{31}{4} + 4 (\frac{- 1}{2}) - 5$

Factor $2$ out of $4$ .

$- \frac{3}{4} + \frac{31}{4} + 2 (2) \frac{- 1}{2} - 5$

Cancel the common factor.

$- \frac{3}{4} + \frac{31}{4} + 2 \cdot 2 \frac{- 1}{2} - 5$

Rewrite the expression.

$- \frac{3}{4} + \frac{31}{4} + 2 \cdot - 1 - 5$

Multiply $2$ by $- 1$ .

$- \frac{3}{4} + \frac{31}{4} - 2 - 5$

Combine fractions.

Combine the numerators over the common denominator.

$- 2 - 5 + \frac{- 3 + 31}{4}$

Add $- 3$ and $31$ .

$- 2 - 5 + \frac{28}{4}$

Find the common denominator.

Write $- 2$ as a fraction with denominator $1$ .

$\frac{- 2}{1} - 5 + \frac{28}{4}$

Multiply $\frac{- 2}{1}$ by $\frac{4}{4}$ .

$\frac{- 2}{1} \cdot \frac{4}{4} - 5 + \frac{28}{4}$

Multiply $\frac{- 2}{1}$ by $\frac{4}{4}$ .

$\frac{- 2 \cdot 4}{4} - 5 + \frac{28}{4}$

Write $- 5$ as a fraction with denominator $1$ .

$\frac{- 2 \cdot 4}{4} + \frac{- 5}{1} + \frac{28}{4}$

Multiply $\frac{- 5}{1}$ by $\frac{4}{4}$ .

$\frac{- 2 \cdot 4}{4} + \frac{- 5}{1} \cdot \frac{4}{4} + \frac{28}{4}$

Multiply $\frac{- 5}{1}$ by $\frac{4}{4}$ .

$\frac{- 2 \cdot 4}{4} + \frac{- 5 \cdot 4}{4} + \frac{28}{4}$

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 4 - 5 \cdot 4 + 28}{4}$

Simplify each term.

Multiply $- 2$ by $4$ .

$\frac{- 8 - 5 \cdot 4 + 28}{4}$

Multiply $- 5$ by $4$ .

$\frac{- 8 - 20 + 28}{4}$

Simplify the expression.

Subtract $20$ from $- 8$ .

$\frac{- 28 + 28}{4}$

Add $- 28$ and $28$ .

$\frac{0}{4}$

Divide $0$ by $4$ .

$0$

Step 5

Since $- \frac{1}{2}$ is a known root, divide the polynomial by $x + \frac{1}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} + 31 x^{2} + 4 x - 5}{x + \frac{1}{2}}$

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.

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0			\xa0
	$6$

Multiply the newest entry in the result $(6)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 3)$ under the next term in the dividend $(31)$ .

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0	$- 3$		\xa0
	$6$

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

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0	$- 3$		\xa0
	$6$	$28$

Multiply the newest entry in the result $(28)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 14)$ under the next term in the dividend $(4)$ .

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0	$- 3$	$- 14$	\xa0
	$6$	$28$

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

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0	$- 3$	$- 14$	\xa0
	$6$	$28$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- \frac{1}{2})$ and place the result of $(5)$ under the next term in the dividend $(- 5)$ .

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0	$- 3$	$- 14$	$5$
	$6$	$28$	$- 10$

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

$- \frac{1}{2}$	$6$	$31$	$4$	$- 5$
	\xa0	$- 3$	$- 14$	$5$
	$6$	$28$	$- 10$	$0$

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

$6 x^{2} + 28 x - 10$

Step 7

Factor $2$ out of $6 x^{2} + 28 x - 10$ .

Factor $2$ out of $6 x^{2}$ .

$(x + \frac{1}{2}) (2 (3 x^{2}) + 28 x - 10)$

Factor $2$ out of $28 x$ .

$(x + \frac{1}{2}) (2 (3 x^{2}) + 2 (14 x) - 10)$

Factor $2$ out of $- 10$ .

$(x + \frac{1}{2}) (2 (3 x^{2}) + 2 (14 x) + 2 (- 5))$

Factor $2$ out of $2 (3 x^{2}) + 2 (14 x)$ .

$(x + \frac{1}{2}) (2 (3 x^{2} + 14 x) + 2 (- 5))$

Factor $2$ out of $2 (3 x^{2} + 14 x) + 2 (- 5)$ .

$(x + \frac{1}{2}) (2 (3 x^{2} + 14 x - 5))$

Step 8

Factor.

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 14$ .

Factor $14$ out of $14 x$ .

$(x + \frac{1}{2}) (2 (3 x^{2} + 14 (x) - 5))$

Rewrite $14$ as $- 1$ plus $15$

$(x + \frac{1}{2}) (2 (3 x^{2} + (- 1 + 15) x - 5))$

Apply the distributive property.

$(x + \frac{1}{2}) (2 (3 x^{2} - 1 x + 15 x - 5))$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + \frac{1}{2}) (2 ((3 x^{2} - 1 x) + 15 x - 5))$

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

$(x + \frac{1}{2}) (2 (x (3 x - 1) + 5 (3 x - 1)))$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(x + \frac{1}{2}) (2 ((3 x - 1) (x + 5)))$

Remove unnecessary parentheses.

$(x + \frac{1}{2}) (2 (3 x - 1) (x + 5))$

Step 9

Factor the left side of the equation.

Factor $6 x^{3} + 31 x^{2} + 4 x - 5$ 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.

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

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

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 5, \pm 0.8\overline{3}, \pm 2.5, \pm 1.\overline{6}$

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Substitute $- 0.5$ into the polynomial.

$6 {(- 0.5)}^{3} + 31 {(- 0.5)}^{2} + 4 \cdot - 0.5 - 5$

Raise $- 0.5$ to the power of $3$ .

$6 \cdot - 0.125 + 31 {(- 0.5)}^{2} + 4 \cdot - 0.5 - 5$

Multiply $6$ by $- 0.125$ .

$- 0.75 + 31 {(- 0.5)}^{2} + 4 \cdot - 0.5 - 5$

Raise $- 0.5$ to the power of $2$ .

$- 0.75 + 31 \cdot 0.25 + 4 \cdot - 0.5 - 5$

Multiply $31$ by $0.25$ .

$- 0.75 + 7.75 + 4 \cdot - 0.5 - 5$

Add $- 0.75$ and $7.75$ .

$7 + 4 \cdot - 0.5 - 5$

Multiply $4$ by $- 0.5$ .

$7 - 2 - 5$

Subtract $2$ from $7$ .

$5 - 5$

Subtract $5$ from $5$ .

$0$

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} + 31 x^{2} + 4 x - 5}{2 x + 1}$

Divide $6 x^{3} + 31 x^{2} + 4 x - 5$ by $2 x + 1$ .

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

$2 x$

$1$

$6 x^{3}$

$31 x^{2}$

$4 x$

$5$

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $2 x$ .

					$3 x^{2}$
	$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			+	$6 x^{3}$	+	$3 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 3 x^{2}$

				$3 x^{2}$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$

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

				$3 x^{2}$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$

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

				$3 x^{2}$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$

Divide the highest order term in the dividend $28 x^{2}$ by the highest order term in divisor $2 x$ .

				$3 x^{2}$	+	$14 x$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$14 x$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					+	$28 x^{2}$	+	$14 x$

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{2} + 14 x$

				$3 x^{2}$	+	$14 x$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$

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

				$3 x^{2}$	+	$14 x$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$
							-	$10 x$

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

				$3 x^{2}$	+	$14 x$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$
							-	$10 x$	-	$5$

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

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$
							-	$10 x$	-	$5$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$
							-	$10 x$	-	$5$
							-	$10 x$	-	$5$

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

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$
							-	$10 x$	-	$5$
							+	$10 x$	+	$5$

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

				$3 x^{2}$	+	$14 x$	-	$5$
$2 x$	+	$1$		$6 x^{3}$	+	$31 x^{2}$	+	$4 x$	-	$5$
			-	$6 x^{3}$	-	$3 x^{2}$
					+	$28 x^{2}$	+	$4 x$
					-	$28 x^{2}$	-	$14 x$
							-	$10 x$	-	$5$
							+	$10 x$	+	$5$
										$0$

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

$3 x^{2} + 14 x - 5$

Write $6 x^{3} + 31 x^{2} + 4 x - 5$ as a set of factors.

$(2 x + 1) (3 x^{2} + 14 x - 5) = 0$

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 14$ .

Factor $14$ out of $14 x$ .

$(2 x + 1) (3 x^{2} + 14 (x) - 5) = 0$

Rewrite $14$ as $- 1$ plus $15$

$(2 x + 1) (3 x^{2} + (- 1 + 15) x - 5) = 0$

Apply the distributive property.

$(2 x + 1) (3 x^{2} - 1 x + 15 x - 5) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x + 1) ((3 x^{2} - 1 x) + 15 x - 5) = 0$

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

$(2 x + 1) (x (3 x - 1) + 5 (3 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(2 x + 1) ((3 x - 1) (x + 5)) = 0$

Remove unnecessary parentheses.

$(2 x + 1) (3 x - 1) (x + 5) = 0$

Step 10

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

$2 x + 1 = 0$

$3 x - 1 = 0$

$x + 5 = 0$

Step 11

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 12

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 13

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 14

The final solution is all the values that make $(2 x + 1) (3 x - 1) (x + 5) = 0$ true.

$x = - \frac{1}{2}, \frac{1}{3}, - 5$

Step 15

Find the Roots/Zeros Using the Rational Roots Test f(x)=6x^3+31x^2+4x-5

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