Factor x^3+7x^2-36

$x^{3} + 7 x^{2} - 36$

Factor $x^{3} + 7 x^{2} - 36$ using the rational roots test.

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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.

$p = \pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6 q = \pm 1$

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

$\pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6$

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

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Substitute $2$ into the polynomial.

$2^{3} + 7 \cdot 2^{2} - 36$

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

$8 + 7 \cdot 2^{2} - 36$

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

$8 + 7 \cdot 4 - 36$

Multiply $7$ by $4$ .

$8 + 28 - 36$

Add $8$ and $28$ .

$36 - 36$

Subtract $36$ from $36$ .

$0$

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.

$\frac{x^{3} + 7 x^{2} - 36}{x - 2}$

Divide $x^{3} + 7 x^{2} - 36$ by $x - 2$ .

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

$x$

$2$

$x^{3}$

$7 x^{2}$

$0 x$

$36$

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			+	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					+	$9 x^{2}$	-	$18 x$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 18 x$

				$x^{2}$	+	$9 x$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$

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

				$x^{2}$	+	$9 x$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$
							+	$18 x$

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

				$x^{2}$	+	$9 x$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$
							+	$18 x$	-	$36$

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

				$x^{2}$	+	$9 x$	+	$18$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$
							+	$18 x$	-	$36$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$	+	$18$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$
							+	$18 x$	-	$36$
							+	$18 x$	-	$36$

The expression needs to be subtracted from the dividend, so change all the signs in $18 x - 36$

				$x^{2}$	+	$9 x$	+	$18$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$
							+	$18 x$	-	$36$
							-	$18 x$	+	$36$

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

				$x^{2}$	+	$9 x$	+	$18$
$x$	-	$2$		$x^{3}$	+	$7 x^{2}$	+	$0 x$	-	$36$
			-	$x^{3}$	+	$2 x^{2}$
					+	$9 x^{2}$	+	$0 x$
					-	$9 x^{2}$	+	$18 x$
							+	$18 x$	-	$36$
							-	$18 x$	+	$36$
										$0$

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

$x^{2} + 9 x + 18$

Write $x^{3} + 7 x^{2} - 36$ as a set of factors.

$(x - 2) (x^{2} + 9 x + 18)$

Factor $x^{2} + 9 x + 18$ using the AC method.

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Factor $x^{2} + 9 x + 18$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$(x - 2) ((x + 3) (x + 6))$

Remove unnecessary parentheses.

$(x - 2) (x + 3) (x + 6)$