Factor x^3-2x^2-23x+60

$x^{3} - 2 x^{2} - 23 x + 60$

Step 1

Factor $x^{3} - 2 x^{2} - 23 x + 60$ 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.

$p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10 q = \pm 1$

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

$\pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

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.

$3^{3} - 2 \cdot 3^{2} - 23 \cdot 3 + 60$

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

$27 - 2 \cdot 3^{2} - 23 \cdot 3 + 60$

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

$27 - 2 \cdot 9 - 23 \cdot 3 + 60$

Multiply $- 2$ by $9$ .

$27 - 18 - 23 \cdot 3 + 60$

Subtract $18$ from $27$ .

$9 - 23 \cdot 3 + 60$

Multiply $- 23$ by $3$ .

$9 - 69 + 60$

Subtract $69$ from $9$ .

$- 60 + 60$

Add $- 60$ and $60$ .

$0$

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.

$\frac{x^{3} - 2 x^{2} - 23 x + 60}{x - 3}$

Divide $x^{3} - 2 x^{2} - 23 x + 60$ 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 $0$ .

$x$

$3$

$x^{3}$

$2 x^{2}$

$23 x$

$60$

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			+	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					+	$x^{2}$	-	$3 x$

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$
							-	$20 x$

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

				$x^{2}$	+	$x$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$
							-	$20 x$	+	$60$

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$
							-	$20 x$	+	$60$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$
							-	$20 x$	+	$60$
							-	$20 x$	+	$60$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x + 60$

				$x^{2}$	+	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$
							-	$20 x$	+	$60$
							+	$20 x$	-	$60$

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

				$x^{2}$	+	$x$	-	$20$
$x$	-	$3$		$x^{3}$	-	$2 x^{2}$	-	$23 x$	+	$60$
			-	$x^{3}$	+	$3 x^{2}$
					+	$x^{2}$	-	$23 x$
					-	$x^{2}$	+	$3 x$
							-	$20 x$	+	$60$
							+	$20 x$	-	$60$
										$0$

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

$x^{2} + x - 20$

Write $x^{3} - 2 x^{2} - 23 x + 60$ as a set of factors.

$(x - 3) (x^{2} + x - 20)$

Step 2

Factor $x^{2} + x - 20$ using the AC method.

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$(x - 3) ((x - 4) (x + 5))$

Remove unnecessary parentheses.

$(x - 3) (x - 4) (x + 5)$

Factor x^3-2x^2-23x+60

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