Factor over the Complex Numbers 11x^4-x^3-33x^2+575x-52

$11 x^{4} - x^{3} - 33 x^{2} + 575 x - 52$

Step 1

Factor $11 x^{4} - x^{3} - 33 x^{2} + 575 x - 52$ 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.

$p = \pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13 q = \pm 1, \pm 11$

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

$\pm 1, \pm 0.\overline{09}, \pm 52, \pm 4.\overline{72}, \pm 2, \pm 0.\overline{18}, \pm 26, \pm 2.\overline{36}, \pm 4, \pm 0.\overline{36}, \pm 13, \pm 1.\overline{18}$

Substitute $0.\overline{09}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{09}$ is a root of the polynomial.

Substitute $0.\overline{09}$ into the polynomial.

$11 \cdot {0.\overline{09}}^{4} - {0.\overline{09}}^{3} - 33 \cdot {0.\overline{09}}^{2} + 575 \cdot 0.\overline{09} - 52$

Raise $0.\overline{09}$ to the power of $4$ .

$11 \cdot 0.0000683 - {0.\overline{09}}^{3} - 33 \cdot {0.\overline{09}}^{2} + 575 \cdot 0.\overline{09} - 52$

Multiply $11$ by $0.0000683$ .

$0.00075131 - {0.\overline{09}}^{3} - 33 \cdot {0.\overline{09}}^{2} + 575 \cdot 0.\overline{09} - 52$

Raise $0.\overline{09}$ to the power of $3$ .

$0.00075131 - 1 \cdot 0.00075131 - 33 \cdot {0.\overline{09}}^{2} + 575 \cdot 0.\overline{09} - 52$

Multiply $- 1$ by $0.00075131$ .

$0.00075131 - 0.00075131 - 33 \cdot {0.\overline{09}}^{2} + 575 \cdot 0.\overline{09} - 52$

Subtract $0.00075131$ from $0.00075131$ .

$0 - 33 \cdot {0.\overline{09}}^{2} + 575 \cdot 0.\overline{09} - 52$

Raise $0.\overline{09}$ to the power of $2$ .

$0 - 33 \cdot 0.00826446 + 575 \cdot 0.\overline{09} - 52$

Multiply $- 33$ by $0.00826446$ .

$0 - 0.27272727 + 575 \cdot 0.\overline{09} - 52$

Subtract $0.27272727$ from $0$ .

$- 0.27272727 + 575 \cdot 0.\overline{09} - 52$

Multiply $575$ by $0.\overline{09}$ .

$- 0.27272727 + 52.27272727 - 52$

Add $- 0.27272727$ and $52.27272727$ .

$52 - 52$

Subtract $52$ from $52$ .

$0$

Since $0.\overline{09}$ is a known root, divide the polynomial by $11 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{11 x^{4} - x^{3} - 33 x^{2} + 575 x - 52}{11 x - 1}$

Divide $11 x^{4} - x^{3} - 33 x^{2} + 575 x - 52$ by $11 x - 1$ .

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

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

Divide the highest order term in the dividend $11 x^{4}$ by the highest order term in divisor $11 x$ .

					$x^{3}$
	$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$

Multiply the new quotient term by the divisor.

				$x^{3}$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			+	$11 x^{4}$	-	$x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $11 x^{4} - x^{3}$

				$x^{3}$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			-	$11 x^{4}$	+	$x^{3}$

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

				$x^{3}$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			-	$11 x^{4}$	+	$x^{3}$
						$0$

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

				$x^{3}$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			-	$11 x^{4}$	+	$x^{3}$
						$0$	-	$33 x^{2}$	+	$575 x$

Divide the highest order term in the dividend $- 33 x^{2}$ by the highest order term in divisor $11 x$ .

$x^{3}$

$0 x^{2}$

$3 x$

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

$11 x^{4}$

$x^{3}$

$0$

$33 x^{2}$

$575 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$3 x$

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

$11 x^{4}$

$x^{3}$

$0$

$33 x^{2}$

$575 x$

$33 x^{2}$

$3 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 33 x^{2} + 3 x$

$x^{3}$

$0 x^{2}$

$3 x$

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

$11 x^{4}$

$x^{3}$

$0$

$33 x^{2}$

$575 x$

$33 x^{2}$

$3 x$

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

$x^{3}$

$0 x^{2}$

$3 x$

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

$11 x^{4}$

$x^{3}$

$0$

$33 x^{2}$

$575 x$

$33 x^{2}$

$3 x$

$572 x$

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

$x^{3}$

$0 x^{2}$

$3 x$

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

$11 x^{4}$

$x^{3}$

$0$

$33 x^{2}$

$575 x$

$33 x^{2}$

$3 x$

$572 x$

$52$

Divide the highest order term in the dividend $572 x$ by the highest order term in divisor $11 x$ .

$x^{3}$

$0 x^{2}$

$3 x$

$52$

$11 x$

$1$

$11 x^{4}$

$x^{3}$

$33 x^{2}$

$575 x$

$52$

$11 x^{4}$

$x^{3}$

$0$

$33 x^{2}$

$575 x$

$33 x^{2}$

$3 x$

$572 x$

$52$

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			-	$11 x^{4}$	+	$x^{3}$
						$0$	-	$33 x^{2}$	+	$575 x$
							+	$33 x^{2}$	-	$3 x$
									+	$572 x$	-	$52$
									+	$572 x$	-	$52$

The expression needs to be subtracted from the dividend, so change all the signs in $572 x - 52$

				$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			-	$11 x^{4}$	+	$x^{3}$
						$0$	-	$33 x^{2}$	+	$575 x$
							+	$33 x^{2}$	-	$3 x$
									+	$572 x$	-	$52$
									-	$572 x$	+	$52$

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

				$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
$11 x$	-	$1$		$11 x^{4}$	-	$x^{3}$	-	$33 x^{2}$	+	$575 x$	-	$52$
			-	$11 x^{4}$	+	$x^{3}$
						$0$	-	$33 x^{2}$	+	$575 x$
							+	$33 x^{2}$	-	$3 x$
									+	$572 x$	-	$52$
									-	$572 x$	+	$52$
												$0$

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

$x^{3} + 0 x^{2} - 3 x + 52$

Write $11 x^{4} - x^{3} - 33 x^{2} + 575 x - 52$ as a set of factors.

$(11 x - 1) (x^{3} + 0 x^{2} - 3 x + 52)$

Step 2

Multiply $0$ by $x^{2}$ .

$(11 x - 1) (x^{3} + 0 - 3 x + 52)$

Step 3

Add $x^{3}$ and $0$ .

$(11 x - 1) (x^{3} - 3 x + 52)$

Step 4

Factor $x^{3} - 3 x + 52$ 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.

$p = \pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13 q = \pm 1$

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

$\pm 1, \pm 52, \pm 2, \pm 26, \pm 4, \pm 13$

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.

${(- 4)}^{3} - 3 \cdot - 4 + 52$

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

$- 64 - 3 \cdot - 4 + 52$

Multiply $- 3$ by $- 4$ .

$- 64 + 12 + 52$

Add $- 64$ and $12$ .

$- 52 + 52$

Add $- 52$ and $52$ .

$0$

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.

$\frac{x^{3} - 3 x + 52}{x + 4}$

Divide $x^{3} - 3 x + 52$ 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 $0$ .

$x$

$4$

$x^{3}$

$0 x^{2}$

$3 x$

$52$

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

					$x^{2}$
	$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			+	$x^{3}$	+	$4 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					-	$4 x^{2}$	-	$16 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 16 x$

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$
							+	$13 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$
							+	$13 x$	+	$52$

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$13$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$
							+	$13 x$	+	$52$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$13$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$
							+	$13 x$	+	$52$
							+	$13 x$	+	$52$

The expression needs to be subtracted from the dividend, so change all the signs in $13 x + 52$

				$x^{2}$	-	$4 x$	+	$13$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$
							+	$13 x$	+	$52$
							-	$13 x$	-	$52$

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

				$x^{2}$	-	$4 x$	+	$13$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$3 x$	+	$52$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$3 x$
					+	$4 x^{2}$	+	$16 x$
							+	$13 x$	+	$52$
							-	$13 x$	-	$52$
										$0$

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

$x^{2} - 4 x + 13$

Write $x^{3} - 3 x + 52$ as a set of factors.

$(11 x - 1) ((x + 4) (x^{2} - 4 x + 13))$

Remove unnecessary parentheses.

$(11 x - 1) (x + 4) (x^{2} - 4 x + 13)$

Factor over the Complex Numbers 11x^4-x^3-33x^2+575x-52

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