Pre-Algebra Examples

$96 z^{5} + 72 z^{3} - 48 z^{2}$

Step 1

Factor $24 z^{2}$ out of $96 z^{5} + 72 z^{3} - 48 z^{2}$ .

Tap for more steps...

Step 1.1

Factor $24 z^{2}$ out of $96 z^{5}$ .

$24 z^{2} (4 z^{3}) + 72 z^{3} - 48 z^{2}$

Step 1.2

Factor $24 z^{2}$ out of $72 z^{3}$ .

$24 z^{2} (4 z^{3}) + 24 z^{2} (3 z) - 48 z^{2}$

Step 1.3

Factor $24 z^{2}$ out of $- 48 z^{2}$ .

$24 z^{2} (4 z^{3}) + 24 z^{2} (3 z) + 24 z^{2} (- 2)$

Step 1.4

Factor $24 z^{2}$ out of $24 z^{2} (4 z^{3}) + 24 z^{2} (3 z)$ .

$24 z^{2} (4 z^{3} + 3 z) + 24 z^{2} (- 2)$

Step 1.5

Factor $24 z^{2}$ out of $24 z^{2} (4 z^{3} + 3 z) + 24 z^{2} (- 2)$ .

$24 z^{2} (4 z^{3} + 3 z - 2)$

Step 2

Factor.

Tap for more steps...

Step 2.1

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

Tap for more steps...

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 4, \pm 2$

Step 2.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 2$

Step 2.1.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Tap for more steps...

Step 2.1.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} + 3 \cdot 0.5 - 2$

Step 2.1.3.2

Raise $0.5$ to the power of $3$ .

$4 \cdot 0.125 + 3 \cdot 0.5 - 2$

Step 2.1.3.3

Multiply $4$ by $0.125$ .

$0.5 + 3 \cdot 0.5 - 2$

Step 2.1.3.4

Multiply $3$ by $0.5$ .

$0.5 + 1.5 - 2$

Step 2.1.3.5

Add $0.5$ and $1.5$ .

$2 - 2$

Step 2.1.3.6

Subtract $2$ from $2$ .

$0$

Step 2.1.4

Since $0.5$ is a known root, divide the polynomial by $2 z - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 2.1.5

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

Tap for more steps...

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 $0$ .

$2 z$

$1$

$4 z^{3}$

$0 z^{2}$

$3 z$

$2$

Step 2.1.5.2

Divide the highest order term in the dividend $4 z^{3}$ by the highest order term in divisor $2 z$ .

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

Step 2.1.5.3

Multiply the new quotient term by the divisor.

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

Step 2.1.5.4

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

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

Step 2.1.5.5

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

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

Step 2.1.5.6

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

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

Step 2.1.5.7

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

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

Step 2.1.5.8

Multiply the new quotient term by the divisor.

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

Step 2.1.5.9

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

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

Step 2.1.5.10

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

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

Step 2.1.5.11

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

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

Step 2.1.5.12

Divide the highest order term in the dividend $4 z$ by the highest order term in divisor $2 z$ .

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

Step 2.1.5.13

Multiply the new quotient term by the divisor.

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

Step 2.1.5.14

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

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

Step 2.1.5.15

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

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

Step 2.1.5.16

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

$2 z^{2} + z + 2$

Step 2.1.6

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

$24 z^{2} ((2 z - 1) (2 z^{2} + z + 2))$

Step 2.2

Remove unnecessary parentheses.

$24 z^{2} (2 z - 1) (2 z^{2} + z + 2)$