Pre-Algebra Examples

$\frac{2 z^{4} - 8 z^{3} + 17 z^{2} - 40 z - 23}{z^{2} - z + 7}$

Step 1

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

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

Step 2

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

							$2 z^{2}$
	$z^{2}$	-	$z$	+	$7$		$2 z^{4}$	-	$8 z^{3}$	+	$17 z^{2}$	-	$40 z$	-	$23$

Step 3

Multiply the new quotient term by the divisor.

$2 z^{2}$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 z^{4} - 2 z^{3} + 14 z^{2}$

$2 z^{2}$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

Step 5

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

$2 z^{2}$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

Step 6

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

$2 z^{2}$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

Step 7

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

$2 z^{2}$

$6 z$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

Step 8

Multiply the new quotient term by the divisor.

$2 z^{2}$

$6 z$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

$6 z^{3}$

$6 z^{2}$

$42 z$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 z^{3} + 6 z^{2} - 42 z$

$2 z^{2}$

$6 z$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

$6 z^{3}$

$6 z^{2}$

$42 z$

Step 10

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

$2 z^{2}$

$6 z$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

$6 z^{3}$

$6 z^{2}$

$42 z$

$3 z^{2}$

$2 z$

Step 11

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

$2 z^{2}$

$6 z$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

$6 z^{3}$

$6 z^{2}$

$42 z$

$3 z^{2}$

$2 z$

$23$

Step 12

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

$2 z^{2}$

$6 z$

$3$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

$6 z^{3}$

$6 z^{2}$

$42 z$

$3 z^{2}$

$2 z$

$23$

Step 13

Multiply the new quotient term by the divisor.

						$2 z^{2}$	-	$6 z$	-	$3$
$z^{2}$	-	$z$	+	$7$		$2 z^{4}$	-	$8 z^{3}$	+	$17 z^{2}$	-	$40 z$	-	$23$
					-	$2 z^{4}$	+	$2 z^{3}$	-	$14 z^{2}$
							-	$6 z^{3}$	+	$3 z^{2}$	-	$40 z$
							+	$6 z^{3}$	-	$6 z^{2}$	+	$42 z$
									-	$3 z^{2}$	+	$2 z$	-	$23$
									-	$3 z^{2}$	+	$3 z$	-	$21$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 z^{2} + 3 z - 21$

$2 z^{2}$

$6 z$

$3$

$z^{2}$

$z$

$7$

$2 z^{4}$

$8 z^{3}$

$17 z^{2}$

$40 z$

$23$

$2 z^{4}$

$2 z^{3}$

$14 z^{2}$

$6 z^{3}$

$3 z^{2}$

$40 z$

$6 z^{3}$

$6 z^{2}$

$42 z$

$3 z^{2}$

$2 z$

$23$

$3 z^{2}$

$3 z$

$21$

Step 15

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

						$2 z^{2}$	-	$6 z$	-	$3$
$z^{2}$	-	$z$	+	$7$		$2 z^{4}$	-	$8 z^{3}$	+	$17 z^{2}$	-	$40 z$	-	$23$
					-	$2 z^{4}$	+	$2 z^{3}$	-	$14 z^{2}$
							-	$6 z^{3}$	+	$3 z^{2}$	-	$40 z$
							+	$6 z^{3}$	-	$6 z^{2}$	+	$42 z$
									-	$3 z^{2}$	+	$2 z$	-	$23$
									+	$3 z^{2}$	-	$3 z$	+	$21$
											-	$z$	-	$2$

Step 16

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

$2 z^{2} - 6 z - 3 + \frac{- z - 2}{z^{2} - z + 7}$