Find the Roots/Zeros Using the Rational Roots Test 3x^4-x^3-36x^2+60x-16

$3 x^{4} - x^{3} - 36 x^{2} + 60 x - 16$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}, \pm 16, \pm \frac{16}{3}$

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.

$3 {(2)}^{4} - {(2)}^{3} - 36 {(2)}^{2} + 60 (2) - 16$

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 $4$ .

$3 \cdot 16 - {(2)}^{3} - 36 {(2)}^{2} + 60 (2) - 16$

Multiply $3$ by $16$ .

$48 - {(2)}^{3} - 36 {(2)}^{2} + 60 (2) - 16$

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

$48 - 1 \cdot 8 - 36 {(2)}^{2} + 60 (2) - 16$

Multiply $- 1$ by $8$ .

$48 - 8 - 36 {(2)}^{2} + 60 (2) - 16$

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

$48 - 8 - 36 \cdot 4 + 60 (2) - 16$

Multiply $- 36$ by $4$ .

$48 - 8 - 144 + 60 (2) - 16$

Multiply $60$ by $2$ .

$48 - 8 - 144 + 120 - 16$

Simplify by adding and subtracting.

Subtract $8$ from $48$ .

$40 - 144 + 120 - 16$

Subtract $144$ from $40$ .

$- 104 + 120 - 16$

Add $- 104$ and $120$ .

$16 - 16$

Subtract $16$ from $16$ .

$0$

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.

$\frac{3 x^{4} - x^{3} - 36 x^{2} + 60 x - 16}{x - 2}$

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.

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0				\xa0

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0				\xa0
	$3$

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

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$			\xa0
	$3$

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

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$			\xa0
	$3$	$5$

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 $(- 36)$ .

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$	$10$		\xa0
	$3$	$5$

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

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$	$10$		\xa0
	$3$	$5$	$- 26$

Multiply the newest entry in the result $(- 26)$ by the divisor $(2)$ and place the result of $(- 52)$ under the next term in the dividend $(60)$ .

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$	$10$	$- 52$	\xa0
	$3$	$5$	$- 26$

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

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$	$10$	$- 52$	\xa0
	$3$	$5$	$- 26$	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(2)$ and place the result of $(16)$ under the next term in the dividend $(- 16)$ .

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$	$10$	$- 52$	$16$
	$3$	$5$	$- 26$	$8$

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

$2$	$3$	$- 1$	$- 36$	$60$	$- 16$
	\xa0	$6$	$10$	$- 52$	$16$
	$3$	$5$	$- 26$	$8$	$0$

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

$3 x^{3} + 5 x^{2} + (- 26) x + 8$

Simplify the quotient polynomial.

$3 x^{3} + 5 x^{2} - 26 x + 8$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $3 x^{3} + 5 x^{2} - 26 x + 8$ 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.

$p = \pm 1, \pm 8, \pm 2, \pm 4 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 8, \pm 2.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 4, \pm 1.\overline{3}$

Substitute $0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{3}$ is a root of the polynomial.

Substitute $0.\overline{3}$ into the polynomial.

$3 \cdot {0.\overline{3}}^{3} + 5 \cdot {0.\overline{3}}^{2} - 26 \cdot 0.\overline{3} + 8$

Raise $0.\overline{3}$ to the power of $3$ .

$3 \cdot 0.\overline{037} + 5 \cdot {0.\overline{3}}^{2} - 26 \cdot 0.\overline{3} + 8$

Multiply $3$ by $0.\overline{037}$ .

$0.\overline{1} + 5 \cdot {0.\overline{3}}^{2} - 26 \cdot 0.\overline{3} + 8$

Raise $0.\overline{3}$ to the power of $2$ .

$0.\overline{1} + 5 \cdot 0.\overline{1} - 26 \cdot 0.\overline{3} + 8$

Multiply $5$ by $0.\overline{1}$ .

$0.\overline{1} + 0.\overline{5} - 26 \cdot 0.\overline{3} + 8$

Add $0.\overline{1}$ and $0.\overline{5}$ .

$0.\overline{6} - 26 \cdot 0.\overline{3} + 8$

Multiply $- 26$ by $0.\overline{3}$ .

$0.\overline{6} - 8.\overline{6} + 8$

Subtract $8.\overline{6}$ from $0.\overline{6}$ .

$- 8 + 8$

Add $- 8$ and $8$ .

$0$

Since $0.\overline{3}$ is a known root, divide the polynomial by $3 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} + 5 x^{2} - 26 x + 8}{3 x - 1}$

Divide $3 x^{3} + 5 x^{2} - 26 x + 8$ by $3 x - 1$ .

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

$3 x$

$1$

$3 x^{3}$

$5 x^{2}$

$26 x$

$8$

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

					$x^{2}$
	$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			+	$3 x^{3}$	-	$x^{2}$

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

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$

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

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$

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

				$x^{2}$	+	$2 x$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					+	$6 x^{2}$	-	$2 x$

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

				$x^{2}$	+	$2 x$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$

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

				$x^{2}$	+	$2 x$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$
							-	$24 x$

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

				$x^{2}$	+	$2 x$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$
							-	$24 x$	+	$8$

Divide the highest order term in the dividend $- 24 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$2 x$	-	$8$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$
							-	$24 x$	+	$8$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$8$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$
							-	$24 x$	+	$8$
							-	$24 x$	+	$8$

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x + 8$

				$x^{2}$	+	$2 x$	-	$8$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$
							-	$24 x$	+	$8$
							+	$24 x$	-	$8$

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

				$x^{2}$	+	$2 x$	-	$8$
$3 x$	-	$1$		$3 x^{3}$	+	$5 x^{2}$	-	$26 x$	+	$8$
			-	$3 x^{3}$	+	$x^{2}$
					+	$6 x^{2}$	-	$26 x$
					-	$6 x^{2}$	+	$2 x$
							-	$24 x$	+	$8$
							+	$24 x$	-	$8$
										$0$

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

$x^{2} + 2 x - 8$

Write $3 x^{3} + 5 x^{2} - 26 x + 8$ as a set of factors.

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

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Write the factored form using these integers.

$(3 x - 1) ((x - 2) (x + 4)) = 0$

Remove unnecessary parentheses.

$(3 x - 1) (x - 2) (x + 4) = 0$

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

$3 x - 1 = 0$

$x - 2 = 0$

$x + 4 = 0$

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 $1$ .

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

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.

$x = 2$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(3 x - 1) (x - 2) (x + 4) = 0$ true.

$x = \frac{1}{3}, 2, - 4$

Step 8

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

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

Step 9

These are the roots (zeros) of the polynomial $3 x^{4} - x^{3} - 36 x^{2} + 60 x - 16$ .

$x = 2, \frac{1}{3}, - 4$

Step 10

Find the Roots/Zeros Using the Rational Roots Test 3x^4-x^3-36x^2+60x-16

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