Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^4+28x^3+102x^2+148x+39

$f (x) = 3 x^{4} + 28 x^{3} + 102 x^{2} + 148 x + 39$

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.

$p = \pm 1, \pm 3, \pm 13, \pm 39$

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm 3, \pm 13, \pm \frac{13}{3}, \pm 39$

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.

$3 {(- 3)}^{4} + 28 {(- 3)}^{3} + 102 {(- 3)}^{2} + 148 (- 3) + 39$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 3$ is a root of the polynomial.

Simplify each term.

Raise $- 3$ to the power of $4$ .

$3 \cdot 81 + 28 {(- 3)}^{3} + 102 {(- 3)}^{2} + 148 (- 3) + 39$

Multiply $3$ by $81$ .

$243 + 28 {(- 3)}^{3} + 102 {(- 3)}^{2} + 148 (- 3) + 39$

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

$243 + 28 \cdot - 27 + 102 {(- 3)}^{2} + 148 (- 3) + 39$

Multiply $28$ by $- 27$ .

$243 - 756 + 102 {(- 3)}^{2} + 148 (- 3) + 39$

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

$243 - 756 + 102 \cdot 9 + 148 (- 3) + 39$

Multiply $102$ by $9$ .

$243 - 756 + 918 + 148 (- 3) + 39$

Multiply $148$ by $- 3$ .

$243 - 756 + 918 - 444 + 39$

Simplify by adding and subtracting.

Subtract $756$ from $243$ .

$- 513 + 918 - 444 + 39$

Add $- 513$ and $918$ .

$405 - 444 + 39$

Subtract $444$ from $405$ .

$- 39 + 39$

Add $- 39$ and $39$ .

$0$

Step 5

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{4} + 28 x^{3} + 102 x^{2} + 148 x + 39}{x + 3}$

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.

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0				\xa0

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

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0				\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(- 3)$ and place the result of $(- 9)$ under the next term in the dividend $(28)$ .

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$			\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.

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$			\xa0
	$3$	$19$

Multiply the newest entry in the result $(19)$ by the divisor $(- 3)$ and place the result of $(- 57)$ under the next term in the dividend $(102)$ .

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$	$- 57$		\xa0
	$3$	$19$

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

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$	$- 57$		\xa0
	$3$	$19$	$45$

Multiply the newest entry in the result $(45)$ by the divisor $(- 3)$ and place the result of $(- 135)$ under the next term in the dividend $(148)$ .

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$	$- 57$	$- 135$	\xa0
	$3$	$19$	$45$

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

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$	$- 57$	$- 135$	\xa0
	$3$	$19$	$45$	$13$

Multiply the newest entry in the result $(13)$ by the divisor $(- 3)$ and place the result of $(- 39)$ under the next term in the dividend $(39)$ .

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$	$- 57$	$- 135$	$- 39$
	$3$	$19$	$45$	$13$

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

$- 3$	$3$	$28$	$102$	$148$	$39$
	\xa0	$- 9$	$- 57$	$- 135$	$- 39$
	$3$	$19$	$45$	$13$	$0$

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

$3 x^{3} + 19 x^{2} + (45) x + 13$

Simplify the quotient polynomial.

$3 x^{3} + 19 x^{2} + 45 x + 13$

Step 7

Solve the equation to find any remaining roots.

Factor $3 x^{3} + 19 x^{2} + 45 x + 13$ 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.

$p = \pm 1, \pm 13 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 13, \pm 4.\overline{3}$

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.

$3 {(- 0.\overline{3})}^{3} + 19 {(- 0.\overline{3})}^{2} + 45 \cdot - 0.\overline{3} + 13$

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

$3 \cdot - 0.\overline{037} + 19 {(- 0.\overline{3})}^{2} + 45 \cdot - 0.\overline{3} + 13$

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

$- 0.\overline{1} + 19 {(- 0.\overline{3})}^{2} + 45 \cdot - 0.\overline{3} + 13$

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

$- 0.\overline{1} + 19 \cdot 0.\overline{1} + 45 \cdot - 0.\overline{3} + 13$

Multiply $19$ by $0.\overline{1}$ .

$- 0.\overline{1} + 2.11111111 + 45 \cdot - 0.\overline{3} + 13$

Add $- 0.\overline{1}$ and $2.11111111$ .

$2 + 45 \cdot - 0.\overline{3} + 13$

Multiply $45$ by $- 0.\overline{3}$ .

$2 - 15 + 13$

Subtract $15$ from $2$ .

$- 13 + 13$

Add $- 13$ and $13$ .

$0$

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.

$\frac{3 x^{3} + 19 x^{2} + 45 x + 13}{3 x + 1}$

Divide $3 x^{3} + 19 x^{2} + 45 x + 13$ 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 $0$ .

$3 x$

$1$

$3 x^{3}$

$19 x^{2}$

$45 x$

$13$

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

					$x^{2}$
	$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			+	$3 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$

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

				$x^{2}$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$

Divide the highest order term in the dividend $18 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$6 x$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					+	$18 x^{2}$	+	$6 x$

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{2} + 6 x$

				$x^{2}$	+	$6 x$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$

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

				$x^{2}$	+	$6 x$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$
							+	$39 x$

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

				$x^{2}$	+	$6 x$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$
							+	$39 x$	+	$13$

Divide the highest order term in the dividend $39 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$6 x$	+	$13$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$
							+	$39 x$	+	$13$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	+	$13$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$
							+	$39 x$	+	$13$
							+	$39 x$	+	$13$

The expression needs to be subtracted from the dividend, so change all the signs in $39 x + 13$

				$x^{2}$	+	$6 x$	+	$13$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$
							+	$39 x$	+	$13$
							-	$39 x$	-	$13$

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

				$x^{2}$	+	$6 x$	+	$13$
$3 x$	+	$1$		$3 x^{3}$	+	$19 x^{2}$	+	$45 x$	+	$13$
			-	$3 x^{3}$	-	$x^{2}$
					+	$18 x^{2}$	+	$45 x$
					-	$18 x^{2}$	-	$6 x$
							+	$39 x$	+	$13$
							-	$39 x$	-	$13$
										$0$

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

$x^{2} + 6 x + 13$

Write $3 x^{3} + 19 x^{2} + 45 x + 13$ as a set of factors.

$(3 x + 1) (x^{2} + 6 x + 13) = 0$

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

$3 x + 1 = 0$

$x^{2} + 6 x + 13 = 0$

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 $x^{2} + 6 x + 13$ equal to $0$ and solve for $x$ .

Set $x^{2} + 6 x + 13$ equal to $0$ .

$x^{2} + 6 x + 13 = 0$

Solve $x^{2} + 6 x + 13 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 6$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{- 6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 4 i}{2}$

Simplify $\frac{- 6 \pm 4 i}{2}$ .

$x = - 3 \pm 2 i$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{- 6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 4 i}{2}$

Simplify $\frac{- 6 \pm 4 i}{2}$ .

$x = - 3 \pm 2 i$

Change the $\pm$ to $+$ .

$x = - 3 + 2 i$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{- 6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 4 i}{2}$

Simplify $\frac{- 6 \pm 4 i}{2}$ .

$x = - 3 \pm 2 i$

Change the $\pm$ to $-$ .

$x = - 3 - 2 i$

The final answer is the combination of both solutions.

$x = - 3 + 2 i, - 3 - 2 i$

The final solution is all the values that make $(3 x + 1) (x^{2} + 6 x + 13) = 0$ true.

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

Step 8

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

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

Step 9

These are the roots (zeros) of the polynomial $3 x^{4} + 28 x^{3} + 102 x^{2} + 148 x + 39$ .

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

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^4+28x^3+102x^2+148x+39

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