Factor over the Complex Numbers 2n^3-9n^2-2n-15

$2 n^{3} - 9 n^{2} - 2 n - 15$

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.

$p = \pm 1, \pm 15, \pm 3, \pm 5 q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm 0.5, \pm 15, \pm 7.5, \pm 3, \pm 1.5, \pm 5, \pm 2.5$

Step 3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

Substitute $5$ into the polynomial.

$2 \cdot 5^{3} - 9 \cdot 5^{2} - 2 \cdot 5 - 15$

Raise $5$ to the power of $3$ .

$2 \cdot 125 - 9 \cdot 5^{2} - 2 \cdot 5 - 15$

Multiply $2$ by $125$ .

$250 - 9 \cdot 5^{2} - 2 \cdot 5 - 15$

Raise $5$ to the power of $2$ .

$250 - 9 \cdot 25 - 2 \cdot 5 - 15$

Multiply $- 9$ by $25$ .

$250 - 225 - 2 \cdot 5 - 15$

Subtract $225$ from $250$ .

$25 - 2 \cdot 5 - 15$

Multiply $- 2$ by $5$ .

$25 - 10 - 15$

Subtract $10$ from $25$ .

$15 - 15$

Subtract $15$ from $15$ .

$0$

Step 4

Since $5$ is a known root, divide the polynomial by $n - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 n^{3} - 9 n^{2} - 2 n - 15}{n - 5}$

Step 5

Divide $2 n^{3} - 9 n^{2} - 2 n - 15$ by $n - 5$ .

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

$n$

$5$

$2 n^{3}$

$9 n^{2}$

$2 n$

$15$

Divide the highest order term in the dividend $2 n^{3}$ by the highest order term in divisor $n$ .

					$2 n^{2}$
	$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$

Multiply the new quotient term by the divisor.

				$2 n^{2}$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			+	$2 n^{3}$	-	$10 n^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 n^{3} - 10 n^{2}$

				$2 n^{2}$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$

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

				$2 n^{2}$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$

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

				$2 n^{2}$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$

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

				$2 n^{2}$	+	$n$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$

Multiply the new quotient term by the divisor.

				$2 n^{2}$	+	$n$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					+	$n^{2}$	-	$5 n$

The expression needs to be subtracted from the dividend, so change all the signs in $n^{2} - 5 n$

				$2 n^{2}$	+	$n$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$

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

				$2 n^{2}$	+	$n$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$
							+	$3 n$

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

				$2 n^{2}$	+	$n$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$
							+	$3 n$	-	$15$

Divide the highest order term in the dividend $3 n$ by the highest order term in divisor $n$ .

				$2 n^{2}$	+	$n$	+	$3$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$
							+	$3 n$	-	$15$

Multiply the new quotient term by the divisor.

				$2 n^{2}$	+	$n$	+	$3$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$
							+	$3 n$	-	$15$
							+	$3 n$	-	$15$

The expression needs to be subtracted from the dividend, so change all the signs in $3 n - 15$

				$2 n^{2}$	+	$n$	+	$3$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$
							+	$3 n$	-	$15$
							-	$3 n$	+	$15$

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

				$2 n^{2}$	+	$n$	+	$3$
$n$	-	$5$		$2 n^{3}$	-	$9 n^{2}$	-	$2 n$	-	$15$
			-	$2 n^{3}$	+	$10 n^{2}$
					+	$n^{2}$	-	$2 n$
					-	$n^{2}$	+	$5 n$
							+	$3 n$	-	$15$
							-	$3 n$	+	$15$
										$0$

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

$2 n^{2} + n + 3$

Step 6

Write $2 n^{3} - 9 n^{2} - 2 n - 15$ as a set of factors.

$(n - 5) (2 n^{2} + n + 3)$

Factor over the Complex Numbers 2n^3-9n^2-2n-15

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