Write as a Set of Linear Factors x^3+9x^2+20x+6

$x^{3} + 9 x^{2} + 20 x + 6$

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.

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

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

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

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} + 9 {(- 3)}^{2} + 20 \cdot - 3 + 6$

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

$- 27 + 9 {(- 3)}^{2} + 20 \cdot - 3 + 6$

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

$- 27 + 9 \cdot 9 + 20 \cdot - 3 + 6$

Multiply $9$ by $9$ .

$- 27 + 81 + 20 \cdot - 3 + 6$

Add $- 27$ and $81$ .

$54 + 20 \cdot - 3 + 6$

Multiply $20$ by $- 3$ .

$54 - 60 + 6$

Subtract $60$ from $54$ .

$- 6 + 6$

Add $- 6$ and $6$ .

$0$

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.

$\frac{x^{3} + 9 x^{2} + 20 x + 6}{x + 3}$

Divide $x^{3} + 9 x^{2} + 20 x + 6$ by $x + 3$ .

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

$x$

$3$

$x^{3}$

$9 x^{2}$

$20 x$

$6$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$

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

				$x^{2}$	+	$6 x$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					+	$6 x^{2}$	+	$18 x$

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

				$x^{2}$	+	$6 x$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$

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

				$x^{2}$	+	$6 x$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$
							+	$2 x$

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

				$x^{2}$	+	$6 x$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$
							+	$2 x$	+	$6$

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$	+	$2$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$
							+	$2 x$	+	$6$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	+	$2$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$
							+	$2 x$	+	$6$
							+	$2 x$	+	$6$

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

				$x^{2}$	+	$6 x$	+	$2$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$
							+	$2 x$	+	$6$
							-	$2 x$	-	$6$

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

				$x^{2}$	+	$6 x$	+	$2$
$x$	+	$3$		$x^{3}$	+	$9 x^{2}$	+	$20 x$	+	$6$
			-	$x^{3}$	-	$3 x^{2}$
					+	$6 x^{2}$	+	$20 x$
					-	$6 x^{2}$	-	$18 x$
							+	$2 x$	+	$6$
							-	$2 x$	-	$6$
										$0$

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

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

Write $x^{3} + 9 x^{2} + 20 x + 6$ as a set of factors.

$(x + 3) (x^{2} + 6 x + 2)$

Write as a Set of Linear Factors x^3+9x^2+20x+6

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