Pre-Algebra Examples

$(30 w^{3} + 8 w^{2} + 29 w + 22) \div (5 w + 3)$

Step 1

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

$5 w$

$3$

$30 w^{3}$

$8 w^{2}$

$29 w$

$22$

Step 2

Divide the highest order term in the dividend $30 w^{3}$ by the highest order term in divisor $5 w$ .

					$6 w^{2}$
	$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$

Step 3

Multiply the new quotient term by the divisor.

				$6 w^{2}$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			+	$30 w^{3}$	+	$18 w^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $30 w^{3} + 18 w^{2}$

				$6 w^{2}$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$

Step 5

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

				$6 w^{2}$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$

Step 6

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

				$6 w^{2}$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$

Step 7

Divide the highest order term in the dividend $- 10 w^{2}$ by the highest order term in divisor $5 w$ .

				$6 w^{2}$	-	$2 w$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$

Step 8

Multiply the new quotient term by the divisor.

				$6 w^{2}$	-	$2 w$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					-	$10 w^{2}$	-	$6 w$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 w^{2} - 6 w$

				$6 w^{2}$	-	$2 w$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$

Step 10

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

				$6 w^{2}$	-	$2 w$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$
							+	$35 w$

Step 11

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

				$6 w^{2}$	-	$2 w$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$
							+	$35 w$	+	$22$

Step 12

Divide the highest order term in the dividend $35 w$ by the highest order term in divisor $5 w$ .

				$6 w^{2}$	-	$2 w$	+	$7$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$
							+	$35 w$	+	$22$

Step 13

Multiply the new quotient term by the divisor.

				$6 w^{2}$	-	$2 w$	+	$7$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$
							+	$35 w$	+	$22$
							+	$35 w$	+	$21$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $35 w + 21$

				$6 w^{2}$	-	$2 w$	+	$7$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$
							+	$35 w$	+	$22$
							-	$35 w$	-	$21$

Step 15

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

				$6 w^{2}$	-	$2 w$	+	$7$
$5 w$	+	$3$		$30 w^{3}$	+	$8 w^{2}$	+	$29 w$	+	$22$
			-	$30 w^{3}$	-	$18 w^{2}$
					-	$10 w^{2}$	+	$29 w$
					+	$10 w^{2}$	+	$6 w$
							+	$35 w$	+	$22$
							-	$35 w$	-	$21$
									+	$1$

Step 16

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

$6 w^{2} - 2 w + 7 + \frac{1}{5 w + 3}$