Find the Roots/Zeros Using the Rational Roots Test 4x^3-12x^2-11x-2

$4 x^{3} - 12 x^{2} - 11 x - 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 2, \pm 4$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2$

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.

$4 {(- \frac{1}{2})}^{3} - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

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 $- \frac{1}{2}$ .

$4 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

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

$4 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

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

$4 (- \frac{1^{3}}{2^{3}}) - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

One to any power is one.

$4 (- \frac{1}{2^{3}}) - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

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

$4 (- \frac{1}{8}) - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

Cancel the common factor of $4$ .

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

$4 (\frac{- 1}{8}) - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

Factor $4$ out of $8$ .

$4 \frac{- 1}{4 (2)} - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

Cancel the common factor.

$4 \frac{- 1}{4 \cdot 2} - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

Rewrite the expression.

$\frac{- 1}{2} - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

Move the negative in front of the fraction.

$- \frac{1}{2} - 12 {(- \frac{1}{2})}^{2} - 11 (- \frac{1}{2}) - 2$

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

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

$- \frac{1}{2} - 12 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) - 11 (- \frac{1}{2}) - 2$

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

$- \frac{1}{2} - 12 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) - 11 (- \frac{1}{2}) - 2$

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

$- \frac{1}{2} - 12 (1 \frac{1^{2}}{2^{2}}) - 11 (- \frac{1}{2}) - 2$

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

$- \frac{1}{2} - 12 \frac{1^{2}}{2^{2}} - 11 (- \frac{1}{2}) - 2$

One to any power is one.

$- \frac{1}{2} - 12 \frac{1}{2^{2}} - 11 (- \frac{1}{2}) - 2$

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

$- \frac{1}{2} - 12 (\frac{1}{4}) - 11 (- \frac{1}{2}) - 2$

Cancel the common factor of $4$ .

Factor $4$ out of $- 12$ .

$- \frac{1}{2} + 4 (- 3) \frac{1}{4} - 11 (- \frac{1}{2}) - 2$

Cancel the common factor.

$- \frac{1}{2} + 4 \cdot - 3 \frac{1}{4} - 11 (- \frac{1}{2}) - 2$

Rewrite the expression.

$- \frac{1}{2} - 3 - 11 (- \frac{1}{2}) - 2$

Multiply $- 11 (- \frac{1}{2})$ .

Multiply $- 1$ by $- 11$ .

$- \frac{1}{2} - 3 + 11 (\frac{1}{2}) - 2$

Combine $11$ and $\frac{1}{2}$ .

$- \frac{1}{2} - 3 + \frac{11}{2} - 2$

Combine fractions.

Combine the numerators over the common denominator.

$- 3 - 2 + \frac{- 1 + 11}{2}$

Add $- 1$ and $11$ .

$- 3 - 2 + \frac{10}{2}$

Find the common denominator.

Write $- 3$ as a fraction with denominator $1$ .

$\frac{- 3}{1} - 2 + \frac{10}{2}$

Multiply $\frac{- 3}{1}$ by $\frac{2}{2}$ .

$\frac{- 3}{1} \cdot \frac{2}{2} - 2 + \frac{10}{2}$

Multiply $\frac{- 3}{1}$ by $\frac{2}{2}$ .

$\frac{- 3 \cdot 2}{2} - 2 + \frac{10}{2}$

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

$\frac{- 3 \cdot 2}{2} + \frac{- 2}{1} + \frac{10}{2}$

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

$\frac{- 3 \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2} + \frac{10}{2}$

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

$\frac{- 3 \cdot 2}{2} + \frac{- 2 \cdot 2}{2} + \frac{10}{2}$

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 2 - 2 \cdot 2 + 10}{2}$

Simplify each term.

Multiply $- 3$ by $2$ .

$\frac{- 6 - 2 \cdot 2 + 10}{2}$

Multiply $- 2$ by $2$ .

$\frac{- 6 - 4 + 10}{2}$

Simplify the expression.

Subtract $4$ from $- 6$ .

$\frac{- 10 + 10}{2}$

Add $- 10$ and $10$ .

$\frac{0}{2}$

Divide $0$ by $2$ .

$0$

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.

$\frac{4 x^{3} - 12 x^{2} - 11 x - 2}{x + \frac{1}{2}}$

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.

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0			\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0			\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 2)$ under the next term in the dividend $(- 12)$ .

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0	$- 2$		\xa0
	$4$

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

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0	$- 2$		\xa0
	$4$	$- 14$

Multiply the newest entry in the result $(- 14)$ by the divisor $(- \frac{1}{2})$ and place the result of $(7)$ under the next term in the dividend $(- 11)$ .

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0	$- 2$	$7$	\xa0
	$4$	$- 14$

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

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0	$- 2$	$7$	\xa0
	$4$	$- 14$	$- 4$

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

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0	$- 2$	$7$	$2$
	$4$	$- 14$	$- 4$

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

$- \frac{1}{2}$	$4$	$- 12$	$- 11$	$- 2$
	\xa0	$- 2$	$7$	$2$
	$4$	$- 14$	$- 4$	$0$

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

$4 x^{2} + - 14 x - 4$

Simplify the quotient polynomial.

$4 x^{2} - 14 x - 4$

Step 7

Factor $2$ out of $4 x^{2} - 14 x - 4$ .

Factor $2$ out of $4 x^{2}$ .

$(x + \frac{1}{2}) (2 (2 x^{2}) - 14 x - 4)$

Factor $2$ out of $- 14 x$ .

$(x + \frac{1}{2}) (2 (2 x^{2}) + 2 (- 7 x) - 4)$

Factor $2$ out of $- 4$ .

$(x + \frac{1}{2}) (2 (2 x^{2}) + 2 (- 7 x) + 2 (- 2))$

Factor $2$ out of $2 (2 x^{2}) + 2 (- 7 x)$ .

$(x + \frac{1}{2}) (2 (2 x^{2} - 7 x) + 2 (- 2))$

Factor $2$ out of $2 (2 x^{2} - 7 x) + 2 (- 2)$ .

$(x + \frac{1}{2}) (2 (2 x^{2} - 7 x - 2))$

Step 8

Factor $4 x^{3} - 12 x^{2} - 11 x - 2$ 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.

$p = \pm 1, \pm 2 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 2$

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.

$4 {(- 0.5)}^{3} - 12 {(- 0.5)}^{2} - 11 \cdot - 0.5 - 2$

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

$4 \cdot - 0.125 - 12 {(- 0.5)}^{2} - 11 \cdot - 0.5 - 2$

Multiply $4$ by $- 0.125$ .

$- 0.5 - 12 {(- 0.5)}^{2} - 11 \cdot - 0.5 - 2$

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

$- 0.5 - 12 \cdot 0.25 - 11 \cdot - 0.5 - 2$

Multiply $- 12$ by $0.25$ .

$- 0.5 - 3 - 11 \cdot - 0.5 - 2$

Subtract $3$ from $- 0.5$ .

$- 3.5 - 11 \cdot - 0.5 - 2$

Multiply $- 11$ by $- 0.5$ .

$- 3.5 + 5.5 - 2$

Add $- 3.5$ and $5.5$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

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.

$\frac{4 x^{3} - 12 x^{2} - 11 x - 2}{2 x + 1}$

Divide $4 x^{3} - 12 x^{2} - 11 x - 2$ 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 $0$ .

$2 x$

$1$

$4 x^{3}$

$12 x^{2}$

$11 x$

$2$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			+	$4 x^{3}$	+	$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 2 x^{2}$

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$

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

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$

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

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$7 x$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$7 x$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	-	$7 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} - 7 x$

				$2 x^{2}$	-	$7 x$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$

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

				$2 x^{2}$	-	$7 x$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$
							-	$4 x$

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

				$2 x^{2}$	-	$7 x$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$
							-	$4 x$	-	$2$

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$7 x$	-	$2$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$
							-	$4 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$7 x$	-	$2$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$
							-	$4 x$	-	$2$
							-	$4 x$	-	$2$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 2$

				$2 x^{2}$	-	$7 x$	-	$2$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$
							-	$4 x$	-	$2$
							+	$4 x$	+	$2$

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

				$2 x^{2}$	-	$7 x$	-	$2$
$2 x$	+	$1$		$4 x^{3}$	-	$12 x^{2}$	-	$11 x$	-	$2$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	+	$7 x$
							-	$4 x$	-	$2$
							+	$4 x$	+	$2$
										$0$

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

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

Write $4 x^{3} - 12 x^{2} - 11 x - 2$ as a set of factors.

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

Step 9

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

$2 x + 1 = 0$

$2 x^{2} - 7 x - 2 = 0$

Step 10

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 11

Set $2 x^{2} - 7 x - 2$ equal to $0$ and solve for $x$ .

Set $2 x^{2} - 7 x - 2$ equal to $0$ .

$2 x^{2} - 7 x - 2 = 0$

Solve $2 x^{2} - 7 x - 2 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 7$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (2 \cdot - 2)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

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

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 2}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 2$ .

Multiply $- 4$ by $2$ .

$x = \frac{7 \pm \sqrt{49 - 8 \cdot - 2}}{2 \cdot 2}$

Multiply $- 8$ by $- 2$ .

$x = \frac{7 \pm \sqrt{49 + 16}}{2 \cdot 2}$

Add $49$ and $16$ .

$x = \frac{7 \pm \sqrt{65}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{7 \pm \sqrt{65}}{4}$

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{65}}{4}, \frac{7 - \sqrt{65}}{4}$

Step 12

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

$x = - \frac{1}{2}, \frac{7 + \sqrt{65}}{4}, \frac{7 - \sqrt{65}}{4}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, \frac{7 + \sqrt{65}}{4}, \frac{7 - \sqrt{65}}{4}$

Decimal Form:

$x = - 0.5, 3.76556443 \dots, - 0.26556443 \dots$

Step 14

Find the Roots/Zeros Using the Rational Roots Test 4x^3-12x^2-11x-2

Related Questions

Support LCMGCF.com by disabling your adblocker.