Factor over the Complex Numbers 3x^4-19x^3-27x^2+359x-116

$3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$

Step 1

Factor $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ 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.

$p = \pm 1, \pm 116, \pm 2, \pm 58, \pm 4, \pm 29 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 116, \pm 38.\overline{6}, \pm 2, \pm 0.\overline{6}, \pm 58, \pm 19.\overline{3}, \pm 4, \pm 1.\overline{3}, \pm 29, \pm 9.\overline{6}$

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.

$3 \cdot {0.\overline{3}}^{4} - 19 \cdot {0.\overline{3}}^{3} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Raise $0.\overline{3}$ to the power of $4$ .

$3 \cdot 0.01234567 - 19 \cdot {0.\overline{3}}^{3} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Multiply $3$ by $0.01234567$ .

$0.\overline{037} - 19 \cdot {0.\overline{3}}^{3} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

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

$0.\overline{037} - 19 \cdot 0.\overline{037} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Multiply $- 19$ by $0.\overline{037}$ .

$0.\overline{037} - 0.\overline{703} - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

Subtract $0.\overline{703}$ from $0.\overline{037}$ .

$- 0.66666666 - 27 \cdot {0.\overline{3}}^{2} + 359 \cdot 0.\overline{3} - 116$

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

$- 0.66666666 - 27 \cdot 0.\overline{1} + 359 \cdot 0.\overline{3} - 116$

Multiply $- 27$ by $0.\overline{1}$ .

$- 0.66666666 - 3 + 359 \cdot 0.\overline{3} - 116$

Subtract $3$ from $- 0.66666666$ .

$- 3.66666666 + 359 \cdot 0.\overline{3} - 116$

Multiply $359$ by $0.\overline{3}$ .

$- 3.66666666 + 119.\overline{6} - 116$

Add $- 3.66666666$ and $119.\overline{6}$ .

$116 - 116$

Subtract $116$ from $116$ .

$0$

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.

$\frac{3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116}{3 x - 1}$

Divide $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ 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 $0$ .

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

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

					$x^{3}$
	$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$

Multiply the new quotient term by the divisor.

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			+	$3 x^{4}$	-	$x^{3}$

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

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$

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

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$

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

				$x^{3}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$

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

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					-	$18 x^{3}$	+	$6 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{3} + 6 x^{2}$

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$

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

				$x^{3}$	-	$6 x^{2}$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$

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

$x^{3}$

$6 x^{2}$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

Divide the highest order term in the dividend $- 33 x^{2}$ by the highest order term in divisor $3 x$ .

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 33 x^{2} + 11 x$

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

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

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

$348 x$

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

$x^{3}$

$6 x^{2}$

$11 x$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

$348 x$

$116$

Divide the highest order term in the dividend $348 x$ by the highest order term in divisor $3 x$ .

$x^{3}$

$6 x^{2}$

$11 x$

$116$

$3 x$

$1$

$3 x^{4}$

$19 x^{3}$

$27 x^{2}$

$359 x$

$116$

$3 x^{4}$

$x^{3}$

$18 x^{3}$

$27 x^{2}$

$18 x^{3}$

$6 x^{2}$

$33 x^{2}$

$359 x$

$33 x^{2}$

$11 x$

$348 x$

$116$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$	+	$359 x$
							+	$33 x^{2}$	-	$11 x$
									+	$348 x$	-	$116$
									+	$348 x$	-	$116$

The expression needs to be subtracted from the dividend, so change all the signs in $348 x - 116$

				$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$	+	$359 x$
							+	$33 x^{2}$	-	$11 x$
									+	$348 x$	-	$116$
									-	$348 x$	+	$116$

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

				$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
$3 x$	-	$1$		$3 x^{4}$	-	$19 x^{3}$	-	$27 x^{2}$	+	$359 x$	-	$116$
			-	$3 x^{4}$	+	$x^{3}$
					-	$18 x^{3}$	-	$27 x^{2}$
					+	$18 x^{3}$	-	$6 x^{2}$
							-	$33 x^{2}$	+	$359 x$
							+	$33 x^{2}$	-	$11 x$
									+	$348 x$	-	$116$
									-	$348 x$	+	$116$
												$0$

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

$x^{3} - 6 x^{2} - 11 x + 116$

Write $3 x^{4} - 19 x^{3} - 27 x^{2} + 359 x - 116$ as a set of factors.

$(3 x - 1) (x^{3} - 6 x^{2} - 11 x + 116)$

Step 2

Factor $x^{3} - 6 x^{2} - 11 x + 116$ 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.

$p = \pm 1, \pm 116, \pm 2, \pm 58, \pm 4, \pm 29 q = \pm 1$

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

$\pm 1, \pm 116, \pm 2, \pm 58, \pm 4, \pm 29$

Substitute $- 4$ and simplify the expression. In this case, the expression is equal to $0$ so $- 4$ is a root of the polynomial.

Substitute $- 4$ into the polynomial.

${(- 4)}^{3} - 6 {(- 4)}^{2} - 11 \cdot - 4 + 116$

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

$- 64 - 6 {(- 4)}^{2} - 11 \cdot - 4 + 116$

Raise $- 4$ to the power of $2$ .

$- 64 - 6 \cdot 16 - 11 \cdot - 4 + 116$

Multiply $- 6$ by $16$ .

$- 64 - 96 - 11 \cdot - 4 + 116$

Subtract $96$ from $- 64$ .

$- 160 - 11 \cdot - 4 + 116$

Multiply $- 11$ by $- 4$ .

$- 160 + 44 + 116$

Add $- 160$ and $44$ .

$- 116 + 116$

Add $- 116$ and $116$ .

$0$

Since $- 4$ is a known root, divide the polynomial by $x + 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 6 x^{2} - 11 x + 116}{x + 4}$

Divide $x^{3} - 6 x^{2} - 11 x + 116$ by $x + 4$ .

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

$x$

$4$

$x^{3}$

$6 x^{2}$

$11 x$

$116$

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

					$x^{2}$
	$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			+	$x^{3}$	+	$4 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					-	$10 x^{2}$	-	$40 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 40 x$

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$

Divide the highest order term in the dividend $29 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$
							+	$29 x$	+	$116$

The expression needs to be subtracted from the dividend, so change all the signs in $29 x + 116$

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$
							-	$29 x$	-	$116$

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

				$x^{2}$	-	$10 x$	+	$29$
$x$	+	$4$		$x^{3}$	-	$6 x^{2}$	-	$11 x$	+	$116$
			-	$x^{3}$	-	$4 x^{2}$
					-	$10 x^{2}$	-	$11 x$
					+	$10 x^{2}$	+	$40 x$
							+	$29 x$	+	$116$
							-	$29 x$	-	$116$
										$0$

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

$x^{2} - 10 x + 29$

Write $x^{3} - 6 x^{2} - 11 x + 116$ as a set of factors.

$(3 x - 1) ((x + 4) (x^{2} - 10 x + 29))$

Remove unnecessary parentheses.

$(3 x - 1) (x + 4) (x^{2} - 10 x + 29)$

Factor over the Complex Numbers 3x^4-19x^3-27x^2+359x-116

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