Basic Math Examples

$2 z^{3} - 16 z - 16$

Step 1

Factor $2$ out of $2 z^{3} - 16 z - 16$ .

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Step 1.1

Factor $2$ out of $2 z^{3}$ .

$2 (z^{3}) - 16 z - 16$

Step 1.2

Factor $2$ out of $- 16 z$ .

$2 (z^{3}) + 2 (- 8 z) - 16$

Step 1.3

Factor $2$ out of $- 16$ .

$2 z^{3} + 2 (- 8 z) + 2 \cdot - 8$

Step 1.4

Factor $2$ out of $2 z^{3} + 2 (- 8 z)$ .

$2 (z^{3} - 8 z) + 2 \cdot - 8$

Step 1.5

Factor $2$ out of $2 (z^{3} - 8 z) + 2 \cdot - 8$ .

$2 (z^{3} - 8 z - 8)$

Step 2

Factor.

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Step 2.1

Factor $z^{3} - 8 z - 8$ using the rational roots test.

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Step 2.1.1

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.

$p = \pm 1, \pm 8, \pm 2, \pm 4$

$q = \pm 1$

Step 2.1.2

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

$\pm 1, \pm 8, \pm 2, \pm 4$

Step 2.1.3

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

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Step 2.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 8 \cdot - 2 - 8$

Step 2.1.3.2

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

$- 8 - 8 \cdot - 2 - 8$

Step 2.1.3.3

Multiply $- 8$ by $- 2$ .

$- 8 + 16 - 8$

Step 2.1.3.4

Add $- 8$ and $16$ .

$8 - 8$

Step 2.1.3.5

Subtract $8$ from $8$ .

$0$

Step 2.1.4

Since $- 2$ is a known root, divide the polynomial by $z + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{z^{3} - 8 z - 8}{z + 2}$

Step 2.1.5

Divide $z^{3} - 8 z - 8$ by $z + 2$ .

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Step 2.1.5.1

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

$z$

$2$

$z^{3}$

$0 z^{2}$

$8 z$

$8$

Step 2.1.5.2

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

					$z^{2}$
	$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$z^{2}$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			+	$z^{3}$	+	$2 z^{2}$

Step 2.1.5.4

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

				$z^{2}$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$

Step 2.1.5.5

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

				$z^{2}$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$

Step 2.1.5.6

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

				$z^{2}$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$

Step 2.1.5.7

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

				$z^{2}$	-	$2 z$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$z^{2}$	-	$2 z$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					-	$2 z^{2}$	-	$4 z$

Step 2.1.5.9

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

				$z^{2}$	-	$2 z$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$

Step 2.1.5.10

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

				$z^{2}$	-	$2 z$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$
							-	$4 z$

Step 2.1.5.11

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

				$z^{2}$	-	$2 z$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$
							-	$4 z$	-	$8$

Step 2.1.5.12

Divide the highest order term in the dividend $- 4 z$ by the highest order term in divisor $z$ .

				$z^{2}$	-	$2 z$	-	$4$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$
							-	$4 z$	-	$8$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$z^{2}$	-	$2 z$	-	$4$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$
							-	$4 z$	-	$8$
							-	$4 z$	-	$8$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 z - 8$

				$z^{2}$	-	$2 z$	-	$4$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$
							-	$4 z$	-	$8$
							+	$4 z$	+	$8$

Step 2.1.5.15

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

				$z^{2}$	-	$2 z$	-	$4$
$z$	+	$2$		$z^{3}$	+	$0 z^{2}$	-	$8 z$	-	$8$
			-	$z^{3}$	-	$2 z^{2}$
					-	$2 z^{2}$	-	$8 z$
					+	$2 z^{2}$	+	$4 z$
							-	$4 z$	-	$8$
							+	$4 z$	+	$8$
										$0$

Step 2.1.5.16

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

$z^{2} - 2 z - 4$

Step 2.1.6

Write $z^{3} - 8 z - 8$ as a set of factors.

$2 ((z + 2) (z^{2} - 2 z - 4))$

Step 2.2

Remove unnecessary parentheses.

$2 (z + 2) (z^{2} - 2 z - 4)$