Write as a Set of Linear Factors x^3-5x^2+12x+18

$x^{3} - 5 x^{2} + 12 x + 18$

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.

$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6 q = \pm 1$

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

$\pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 5 {(- 1)}^{2} + 12 \cdot - 1 + 18$

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

$- 1 - 5 {(- 1)}^{2} + 12 \cdot - 1 + 18$

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

$- 1 - 5 \cdot 1 + 12 \cdot - 1 + 18$

Multiply $- 5$ by $1$ .

$- 1 - 5 + 12 \cdot - 1 + 18$

Subtract $5$ from $- 1$ .

$- 6 + 12 \cdot - 1 + 18$

Multiply $12$ by $- 1$ .

$- 6 - 12 + 18$

Subtract $12$ from $- 6$ .

$- 18 + 18$

Add $- 18$ and $18$ .

$0$

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.

$\frac{x^{3} - 5 x^{2} + 12 x + 18}{x + 1}$

Divide $x^{3} - 5 x^{2} + 12 x + 18$ by $x + 1$ .

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

$x$

$1$

$x^{3}$

$5 x^{2}$

$12 x$

$18$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					-	$6 x^{2}$	-	$6 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$
							+	$18 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$
							+	$18 x$	+	$18$

Divide the highest order term in the dividend $18 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$18$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$
							+	$18 x$	+	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$18$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$
							+	$18 x$	+	$18$
							+	$18 x$	+	$18$

The expression needs to be subtracted from the dividend, so change all the signs in $18 x + 18$

				$x^{2}$	-	$6 x$	+	$18$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$
							+	$18 x$	+	$18$
							-	$18 x$	-	$18$

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

				$x^{2}$	-	$6 x$	+	$18$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$12 x$	+	$18$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$12 x$
					+	$6 x^{2}$	+	$6 x$
							+	$18 x$	+	$18$
							-	$18 x$	-	$18$
										$0$

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

$x^{2} - 6 x + 18$

Write $x^{3} - 5 x^{2} + 12 x + 18$ as a set of factors.

$(x + 1) (x^{2} - 6 x + 18)$

Write as a Set of Linear Factors x^3-5x^2+12x+18

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