Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^4-23x^3+7x^2+47x+14

$f (x) = 3 x^{4} - 23 x^{3} + 7 x^{2} + 47 x + 14$

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.

$p = \pm 1, \pm 2, \pm 7, \pm 14$

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 7, \pm \frac{7}{3}, \pm 14, \pm \frac{14}{3}$

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.

$3 {(- 1)}^{4} - 23 {(- 1)}^{3} + 7 {(- 1)}^{2} + 47 (- 1) + 14$

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 $4$ .

$3 \cdot 1 - 23 {(- 1)}^{3} + 7 {(- 1)}^{2} + 47 (- 1) + 14$

Multiply $3$ by $1$ .

$3 - 23 {(- 1)}^{3} + 7 {(- 1)}^{2} + 47 (- 1) + 14$

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

$3 - 23 \cdot - 1 + 7 {(- 1)}^{2} + 47 (- 1) + 14$

Multiply $- 23$ by $- 1$ .

$3 + 23 + 7 {(- 1)}^{2} + 47 (- 1) + 14$

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

$3 + 23 + 7 \cdot 1 + 47 (- 1) + 14$

Multiply $7$ by $1$ .

$3 + 23 + 7 + 47 (- 1) + 14$

Multiply $47$ by $- 1$ .

$3 + 23 + 7 - 47 + 14$

Simplify by adding and subtracting.

Add $3$ and $23$ .

$26 + 7 - 47 + 14$

Add $26$ and $7$ .

$33 - 47 + 14$

Subtract $47$ from $33$ .

$- 14 + 14$

Add $- 14$ and $14$ .

$0$

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.

$\frac{3 x^{4} - 23 x^{3} + 7 x^{2} + 47 x + 14}{x + 1}$

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.

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0				\xa0

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

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0				\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(- 1)$ and place the result of $(- 3)$ under the next term in the dividend $(- 23)$ .

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$			\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.

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$			\xa0
	$3$	$- 26$

Multiply the newest entry in the result $(- 26)$ by the divisor $(- 1)$ and place the result of $(26)$ under the next term in the dividend $(7)$ .

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$	$26$		\xa0
	$3$	$- 26$

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

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$	$26$		\xa0
	$3$	$- 26$	$33$

Multiply the newest entry in the result $(33)$ by the divisor $(- 1)$ and place the result of $(- 33)$ under the next term in the dividend $(47)$ .

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$	$26$	$- 33$	\xa0
	$3$	$- 26$	$33$

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

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$	$26$	$- 33$	\xa0
	$3$	$- 26$	$33$	$14$

Multiply the newest entry in the result $(14)$ by the divisor $(- 1)$ and place the result of $(- 14)$ under the next term in the dividend $(14)$ .

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$	$26$	$- 33$	$- 14$
	$3$	$- 26$	$33$	$14$

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

$- 1$	$3$	$- 23$	$7$	$47$	$14$
	\xa0	$- 3$	$26$	$- 33$	$- 14$
	$3$	$- 26$	$33$	$14$	$0$

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

$3 x^{3} + - 26 x^{2} + (33) x + 14$

Simplify the quotient polynomial.

$3 x^{3} - 26 x^{2} + 33 x + 14$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $3 x^{3} - 26 x^{2} + 33 x + 14$ 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.

$p = \pm 1, \pm 14, \pm 2, \pm 7 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 14, \pm 4.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 7, \pm 2.\overline{3}$

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.

$3 {(- 0.\overline{3})}^{3} - 26 {(- 0.\overline{3})}^{2} + 33 \cdot - 0.\overline{3} + 14$

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

$3 \cdot - 0.\overline{037} - 26 {(- 0.\overline{3})}^{2} + 33 \cdot - 0.\overline{3} + 14$

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

$- 0.\overline{1} - 26 {(- 0.\overline{3})}^{2} + 33 \cdot - 0.\overline{3} + 14$

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

$- 0.\overline{1} - 26 \cdot 0.\overline{1} + 33 \cdot - 0.\overline{3} + 14$

Multiply $- 26$ by $0.\overline{1}$ .

$- 0.\overline{1} - 2.\overline{8} + 33 \cdot - 0.\overline{3} + 14$

Subtract $2.\overline{8}$ from $- 0.\overline{1}$ .

$- 3 + 33 \cdot - 0.\overline{3} + 14$

Multiply $33$ by $- 0.\overline{3}$ .

$- 3 - 11 + 14$

Subtract $11$ from $- 3$ .

$- 14 + 14$

Add $- 14$ and $14$ .

$0$

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.

$\frac{3 x^{3} - 26 x^{2} + 33 x + 14}{3 x + 1}$

Divide $3 x^{3} - 26 x^{2} + 33 x + 14$ 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 $0$ .

$3 x$

$1$

$3 x^{3}$

$26 x^{2}$

$33 x$

$14$

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

					$x^{2}$
	$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			+	$3 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$

Divide the highest order term in the dividend $- 27 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$9 x$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					-	$27 x^{2}$	-	$9 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 27 x^{2} - 9 x$

				$x^{2}$	-	$9 x$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$

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

				$x^{2}$	-	$9 x$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$
							+	$42 x$

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

				$x^{2}$	-	$9 x$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$
							+	$42 x$	+	$14$

Divide the highest order term in the dividend $42 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$9 x$	+	$14$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$
							+	$42 x$	+	$14$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$	+	$14$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$
							+	$42 x$	+	$14$
							+	$42 x$	+	$14$

The expression needs to be subtracted from the dividend, so change all the signs in $42 x + 14$

				$x^{2}$	-	$9 x$	+	$14$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$
							+	$42 x$	+	$14$
							-	$42 x$	-	$14$

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

				$x^{2}$	-	$9 x$	+	$14$
$3 x$	+	$1$		$3 x^{3}$	-	$26 x^{2}$	+	$33 x$	+	$14$
			-	$3 x^{3}$	-	$x^{2}$
					-	$27 x^{2}$	+	$33 x$
					+	$27 x^{2}$	+	$9 x$
							+	$42 x$	+	$14$
							-	$42 x$	-	$14$
										$0$

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

$x^{2} - 9 x + 14$

Write $3 x^{3} - 26 x^{2} + 33 x + 14$ as a set of factors.

$(3 x + 1) (x^{2} - 9 x + 14) = 0$

Factor $x^{2} - 9 x + 14$ 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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Write the factored form using these integers.

$(3 x + 1) ((x - 7) (x - 2)) = 0$

Remove unnecessary parentheses.

$(3 x + 1) (x - 7) (x - 2) = 0$

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

$3 x + 1 = 0$

$x - 7 = 0$

$x - 2 = 0$

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 $x$ .

Subtract $1$ from 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 $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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.

$x = 2$

The final solution is all the values that make $(3 x + 1) (x - 7) (x - 2) = 0$ true.

$x = - \frac{1}{3}, 7, 2$

Step 8

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

$(x + 1) (x + \frac{1}{3}) (x - 7) (x - 2)$

Step 9

These are the roots (zeros) of the polynomial $3 x^{4} - 23 x^{3} + 7 x^{2} + 47 x + 14$ .

$x = - 1, - \frac{1}{3}, 7, 2$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^4-23x^3+7x^2+47x+14

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