Precalculus Examples

$x^{3} - x^{2} - 32 x + 60$

Step 1

Set $x^{3} - x^{2} - 32 x + 60$ equal to $0$ .

$x^{3} - x^{2} - 32 x + 60 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Factor the left side of the equation.

Tap for more steps...

Step 2.1.1

Factor $x^{3} - x^{2} - 32 x + 60$ using the rational roots test.

Tap for more steps...

Step 2.1.1.1

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.

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

$q = \pm 1$

Step 2.1.1.2

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

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

Step 2.1.1.3

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

Tap for more steps...

Step 2.1.1.3.1

Substitute $2$ into the polynomial.

$2^{3} - 2^{2} - 32 \cdot 2 + 60$

Step 2.1.1.3.2

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

$8 - 2^{2} - 32 \cdot 2 + 60$

Step 2.1.1.3.3

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

$8 - 1 \cdot 4 - 32 \cdot 2 + 60$

Step 2.1.1.3.4

Multiply $- 1$ by $4$ .

$8 - 4 - 32 \cdot 2 + 60$

Step 2.1.1.3.5

Subtract $4$ from $8$ .

$4 - 32 \cdot 2 + 60$

Step 2.1.1.3.6

Multiply $- 32$ by $2$ .

$4 - 64 + 60$

Step 2.1.1.3.7

Subtract $64$ from $4$ .

$- 60 + 60$

Step 2.1.1.3.8

Add $- 60$ and $60$ .

$0$

Step 2.1.1.4

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.

$\frac{x^{3} - x^{2} - 32 x + 60}{x - 2}$

Step 2.1.1.5

Divide $x^{3} - x^{2} - 32 x + 60$ by $x - 2$ .

Tap for more steps...

Step 2.1.1.5.1

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

$x$

$2$

$x^{3}$

$x^{2}$

$32 x$

$60$

Step 2.1.1.5.2

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.1.1.5.4

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.1.1.5.5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$

Step 2.1.1.5.6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$

Step 2.1.1.5.7

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

				$x^{2}$	+	$x$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					+	$x^{2}$	-	$2 x$

Step 2.1.1.5.9

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

				$x^{2}$	+	$x$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$

Step 2.1.1.5.10

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

				$x^{2}$	+	$x$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$
							-	$30 x$

Step 2.1.1.5.11

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

				$x^{2}$	+	$x$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$
							-	$30 x$	+	$60$

Step 2.1.1.5.12

Divide the highest order term in the dividend $- 30 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$x$	-	$30$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$
							-	$30 x$	+	$60$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$30$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$
							-	$30 x$	+	$60$
							-	$30 x$	+	$60$

Step 2.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 x + 60$

				$x^{2}$	+	$x$	-	$30$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$
							-	$30 x$	+	$60$
							+	$30 x$	-	$60$

Step 2.1.1.5.15

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

				$x^{2}$	+	$x$	-	$30$
$x$	-	$2$		$x^{3}$	-	$x^{2}$	-	$32 x$	+	$60$
			-	$x^{3}$	+	$2 x^{2}$
					+	$x^{2}$	-	$32 x$
					-	$x^{2}$	+	$2 x$
							-	$30 x$	+	$60$
							+	$30 x$	-	$60$
										$0$

Step 2.1.1.5.16

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

$x^{2} + x - 30$

Step 2.1.1.6

Write $x^{3} - x^{2} - 32 x + 60$ as a set of factors.

$(x - 2) (x^{2} + x - 30) = 0$

Step 2.1.2

Factor $x^{2} + x - 30$ using the AC method.

Tap for more steps...

Step 2.1.2.1

Factor $x^{2} + x - 30$ using the AC method.

Tap for more steps...

Step 2.1.2.1.1

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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 2.1.2.1.2

Write the factored form using these integers.

$(x - 2) ((x - 5) (x + 6)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(x - 2) (x - 5) (x + 6) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x - 5 = 0$

$x + 6 = 0$

Step 2.3

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.4

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.5

Set $x + 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.5.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.6

The final solution is all the values that make $(x - 2) (x - 5) (x + 6) = 0$ true.

$x = 2, 5, - 6$

Step 3