Factor over the Complex Numbers x^3-12x^2+49x-58

$x^{3} - 12 x^{2} + 49 x - 58$

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.

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

Step 2

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

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

Step 3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$2^{3} - 12 \cdot 2^{2} + 49 \cdot 2 - 58$

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

$8 - 12 \cdot 2^{2} + 49 \cdot 2 - 58$

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

$8 - 12 \cdot 4 + 49 \cdot 2 - 58$

Multiply $- 12$ by $4$ .

$8 - 48 + 49 \cdot 2 - 58$

Subtract $48$ from $8$ .

$- 40 + 49 \cdot 2 - 58$

Multiply $49$ by $2$ .

$- 40 + 98 - 58$

Add $- 40$ and $98$ .

$58 - 58$

Subtract $58$ from $58$ .

$0$

Step 4

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.

$\frac{x^{3} - 12 x^{2} + 49 x - 58}{x - 2}$

Step 5

Divide $x^{3} - 12 x^{2} + 49 x - 58$ by $x - 2$ .

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

$x$

$2$

$x^{3}$

$12 x^{2}$

$49 x$

$58$

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			+	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$

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

				$x^{2}$	-	$10 x$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					-	$10 x^{2}$	+	$20 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} + 20 x$

				$x^{2}$	-	$10 x$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$

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

				$x^{2}$	-	$10 x$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$
							+	$29 x$

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

				$x^{2}$	-	$10 x$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$
							+	$29 x$	-	$58$

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

				$x^{2}$	-	$10 x$	+	$29$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$
							+	$29 x$	-	$58$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$29$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$
							+	$29 x$	-	$58$
							+	$29 x$	-	$58$

The expression needs to be subtracted from the dividend, so change all the signs in $29 x - 58$

				$x^{2}$	-	$10 x$	+	$29$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$
							+	$29 x$	-	$58$
							-	$29 x$	+	$58$

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

				$x^{2}$	-	$10 x$	+	$29$
$x$	-	$2$		$x^{3}$	-	$12 x^{2}$	+	$49 x$	-	$58$
			-	$x^{3}$	+	$2 x^{2}$
					-	$10 x^{2}$	+	$49 x$
					+	$10 x^{2}$	-	$20 x$
							+	$29 x$	-	$58$
							-	$29 x$	+	$58$
										$0$

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

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

Step 6

Write $x^{3} - 12 x^{2} + 49 x - 58$ as a set of factors.

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

Factor over the Complex Numbers x^3-12x^2+49x-58

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