Factor 6n^4-24n^3+18n

$6 n^{4} - 24 n^{3} + 18 n$

Step 1

Factor $6 n$ out of $6 n^{4} - 24 n^{3} + 18 n$ .

Factor $6 n$ out of $6 n^{4}$ .

$6 n (n^{3}) - 24 n^{3} + 18 n$

Factor $6 n$ out of $- 24 n^{3}$ .

$6 n (n^{3}) + 6 n (- 4 n^{2}) + 18 n$

Factor $6 n$ out of $18 n$ .

$6 n (n^{3}) + 6 n (- 4 n^{2}) + 6 n (3)$

Factor $6 n$ out of $6 n (n^{3}) + 6 n (- 4 n^{2})$ .

$6 n (n^{3} - 4 n^{2}) + 6 n (3)$

Factor $6 n$ out of $6 n (n^{3} - 4 n^{2}) + 6 n (3)$ .

$6 n (n^{3} - 4 n^{2} + 3)$

Step 2

Factor.

Factor $n^{3} - 4 n^{2} + 3$ using the rational roots test.

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.

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

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

$\pm 1, \pm 3$

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.

$1^{3} - 4 \cdot 1^{2} + 3$

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

$1 - 4 \cdot 1^{2} + 3$

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

$1 - 4 \cdot 1 + 3$

Multiply $- 4$ by $1$ .

$1 - 4 + 3$

Subtract $4$ from $1$ .

$- 3 + 3$

Add $- 3$ and $3$ .

$0$

Since $1$ is a known root, divide the polynomial by $n - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{n^{3} - 4 n^{2} + 3}{n - 1}$

Divide $n^{3} - 4 n^{2} + 3$ by $n - 1$ .

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

$n$

$1$

$n^{3}$

$4 n^{2}$

$0 n$

$3$

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

					$n^{2}$
	$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$

Multiply the new quotient term by the divisor.

				$n^{2}$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			+	$n^{3}$	-	$n^{2}$

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

				$n^{2}$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$

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

				$n^{2}$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$

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

				$n^{2}$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$

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

				$n^{2}$	-	$3 n$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$

Multiply the new quotient term by the divisor.

				$n^{2}$	-	$3 n$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					-	$3 n^{2}$	+	$3 n$

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

				$n^{2}$	-	$3 n$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$

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

				$n^{2}$	-	$3 n$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$
							-	$3 n$

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

				$n^{2}$	-	$3 n$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$
							-	$3 n$	+	$3$

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

				$n^{2}$	-	$3 n$	-	$3$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$
							-	$3 n$	+	$3$

Multiply the new quotient term by the divisor.

				$n^{2}$	-	$3 n$	-	$3$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$
							-	$3 n$	+	$3$
							-	$3 n$	+	$3$

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

				$n^{2}$	-	$3 n$	-	$3$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$
							-	$3 n$	+	$3$
							+	$3 n$	-	$3$

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

				$n^{2}$	-	$3 n$	-	$3$
$n$	-	$1$		$n^{3}$	-	$4 n^{2}$	+	$0 n$	+	$3$
			-	$n^{3}$	+	$n^{2}$
					-	$3 n^{2}$	+	$0 n$
					+	$3 n^{2}$	-	$3 n$
							-	$3 n$	+	$3$
							+	$3 n$	-	$3$
										$0$

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

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

Write $n^{3} - 4 n^{2} + 3$ as a set of factors.

$6 n ((n - 1) (n^{2} - 3 n - 3))$

Remove unnecessary parentheses.

$6 n (n - 1) (n^{2} - 3 n - 3)$

Factor 6n^4-24n^3+18n

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