Precalculus Examples

$x^{3} - 48 x - 128$

Step 1

Factor $x^{3} - 48 x - 128$ using the rational roots test.

Tap for more steps...

Step 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.

$p = \pm 1, \pm 128, \pm 2, \pm 64, \pm 4, \pm 32, \pm 8, \pm 16$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 128, \pm 2, \pm 64, \pm 4, \pm 32, \pm 8, \pm 16$

Step 1.3

Substitute $- 4$ and simplify the expression. In this case, the expression is equal to $0$ so $- 4$ is a root of the polynomial.

Tap for more steps...

Step 1.3.1

Substitute $- 4$ into the polynomial.

${(- 4)}^{3} - 48 \cdot - 4 - 128$

Step 1.3.2

Raise $- 4$ to the power of $3$ .

$- 64 - 48 \cdot - 4 - 128$

Step 1.3.3

Multiply $- 48$ by $- 4$ .

$- 64 + 192 - 128$

Step 1.3.4

Add $- 64$ and $192$ .

$128 - 128$

Step 1.3.5

Subtract $128$ from $128$ .

$0$

Step 1.4

Since $- 4$ is a known root, divide the polynomial by $x + 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 48 x - 128}{x + 4}$

Step 1.5

Divide $x^{3} - 48 x - 128$ by $x + 4$ .

Tap for more steps...

Step 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 $0$ .

$x$

$4$

$x^{3}$

$0 x^{2}$

$48 x$

$128$

Step 1.5.2

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

					$x^{2}$
	$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			+	$x^{3}$	+	$4 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 4 x^{2}$

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$

Step 1.5.5

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$

Step 1.5.6

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

				$x^{2}$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$

Step 1.5.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					-	$4 x^{2}$	-	$16 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 16 x$

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$

Step 1.5.10

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

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$
							-	$32 x$

Step 1.5.11

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

				$x^{2}$	-	$4 x$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$
							-	$32 x$	-	$128$

Step 1.5.12

Divide the highest order term in the dividend $- 32 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	-	$32$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$
							-	$32 x$	-	$128$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	-	$32$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$
							-	$32 x$	-	$128$
							-	$32 x$	-	$128$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 32 x - 128$

				$x^{2}$	-	$4 x$	-	$32$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$
							-	$32 x$	-	$128$
							+	$32 x$	+	$128$

Step 1.5.15

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

				$x^{2}$	-	$4 x$	-	$32$
$x$	+	$4$		$x^{3}$	+	$0 x^{2}$	-	$48 x$	-	$128$
			-	$x^{3}$	-	$4 x^{2}$
					-	$4 x^{2}$	-	$48 x$
					+	$4 x^{2}$	+	$16 x$
							-	$32 x$	-	$128$
							+	$32 x$	+	$128$
										$0$

Step 1.5.16

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

$x^{2} - 4 x - 32$

Step 1.6

Write $x^{3} - 48 x - 128$ as a set of factors.

$(x + 4) (x^{2} - 4 x - 32)$

Step 2

Factor $x^{2} - 4 x - 32$ using the AC method.

Tap for more steps...

Step 2.1

Factor $x^{2} - 4 x - 32$ using the AC method.

Tap for more steps...

Step 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 $- 32$ and whose sum is $- 4$ .

$- 8, 4$

Step 2.1.2

Write the factored form using these integers.

$(x + 4) ((x - 8) (x + 4))$

Step 2.2

Remove unnecessary parentheses.

$(x + 4) (x - 8) (x + 4)$