Algebra Examples

$(- 6 w^{3} z^{4} - 3 w^{2} z^{5} + 4 w + 5 z) \div (2 w^{2} z)$

Step 1

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

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

Step 2

Divide the highest order term in the dividend $- 6 z^{4} w^{3}$ by the highest order term in divisor $2 w^{2} z$ .

$3 z^{3} w$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

Step 3

Multiply the new quotient term by the divisor.

$3 z^{3} w$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 z^{4} w^{3} + 0 + 0$

$3 z^{3} w$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

Step 5

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

$3 z^{3} w$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

$3 z^{5} w^{2}$

$4 w$

Step 6

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

$3 z^{3} w$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

$3 z^{5} w^{2}$

$4 w$

$5 z$

Step 7

Divide the highest order term in the dividend $- 3 z^{5} w^{2}$ by the highest order term in divisor $2 w^{2} z$ .

$3 z^{3} w$

$\frac{3 z^{4}}{2}$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

$3 z^{5} w^{2}$

$4 w$

$5 z$

Step 8

Multiply the new quotient term by the divisor.

$3 z^{3} w$

$\frac{3 z^{4}}{2}$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$3 w^{2} z^{5}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 w^{2} z^{5} + 0 + 0$

					-	$3 z^{3} w$	-	$\frac{3 z^{4}}{2}$
$2 w^{2} z$	+	$0 w$	+	$0$	-	$6 z^{4} w^{3}$	-	$3 z^{5} w^{2}$	+	$4 w$	+	$5 z$
					+	$6 z^{4} w^{3}$	-	$0$	-	$0$
							-	$3 z^{5} w^{2}$	+	$4 w$	+	$5 z$
							+	$3 w^{2} z^{5}$	-	$0$	-	$0$

Step 10

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

$3 z^{3} w$

$\frac{3 z^{4}}{2}$

$2 w^{2} z$

$0 w$

$0$

$6 z^{4} w^{3}$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$6 z^{4} w^{3}$

$0$

$3 z^{5} w^{2}$

$4 w$

$5 z$

$3 w^{2} z^{5}$

$0$

$4 w$

$5 z$

Step 11

The final answer is the quotient plus the remainder over the divisor.

$- 3 z^{3} w - \frac{3 z^{4}}{2} + \frac{4 w + 5 z}{2 w^{2} z}$