Precalculus Examples

$\frac{6 a^{4} b^{3} + 3 a^{2} b^{2} - 12 a^{2} b}{3 a b}$

Step 1

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

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

Step 2

Divide the highest order term in the dividend $6 a^{4} b^{3}$ by the highest order term in divisor $3 a b$ .

$2 a^{3} b^{2}$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 a^{3} b^{2}$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

Step 4

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

$2 a^{3} b^{2}$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

Step 5

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

$2 a^{3} b^{2}$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

Step 6

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

$2 a^{3} b^{2}$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

Step 7

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

$2 a^{3} b^{2}$

$0 a^{2}$

$a b$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

Step 8

Multiply the new quotient term by the divisor.

$2 a^{3} b^{2}$

$0 a^{2}$

$a b$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

$3 a^{2} b^{2}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 a^{2} b^{2} + 0$

$2 a^{3} b^{2}$

$0 a^{2}$

$a b$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

$3 a^{2} b^{2}$

$0$

Step 10

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

$2 a^{3} b^{2}$

$0 a^{2}$

$a b$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

$3 a^{2} b^{2}$

$0$

$12 a^{2} b$

Step 11

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

$2 a^{3} b^{2}$

$0 a^{2}$

$a b$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

$3 a^{2} b^{2}$

$0$

$12 a^{2} b$

$0$

Step 12

Divide the highest order term in the dividend $- 12 a^{2} b$ by the highest order term in divisor $3 a b$ .

$2 a^{3} b^{2}$

$0 a^{2}$

$a b$

$4 a$

$3 a b$

$0$

$6 a^{4} b^{3}$

$0 a^{3}$

$3 a^{2} b^{2}$

$12 a^{2} b$

$0$

$6 a^{4} b^{3}$

$0$

$3 a^{2} b^{2}$

$12 a^{2} b$

$3 a^{2} b^{2}$

$0$

$12 a^{2} b$

$0$

Step 13

Multiply the new quotient term by the divisor.

				$2 a^{3} b^{2}$	+	$0 a^{2}$	+	$a b$	-	$4 a$
$3 a b$	+	$0$		$6 a^{4} b^{3}$	+	$0 a^{3}$	+	$3 a^{2} b^{2}$	-	$12 a^{2} b$	+	$0$
			-	$6 a^{4} b^{3}$	-	$0$
						$0$	+	$3 a^{2} b^{2}$	-	$12 a^{2} b$
							-	$3 a^{2} b^{2}$	-	$0$
									-	$12 a^{2} b$	+	$0$
									-	$12 a^{2} b$	+	$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 a^{2} b + 0$

				$2 a^{3} b^{2}$	+	$0 a^{2}$	+	$a b$	-	$4 a$
$3 a b$	+	$0$		$6 a^{4} b^{3}$	+	$0 a^{3}$	+	$3 a^{2} b^{2}$	-	$12 a^{2} b$	+	$0$
			-	$6 a^{4} b^{3}$	-	$0$
						$0$	+	$3 a^{2} b^{2}$	-	$12 a^{2} b$
							-	$3 a^{2} b^{2}$	-	$0$
									-	$12 a^{2} b$	+	$0$
									+	$12 a^{2} b$	-	$0$

Step 15

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

				$2 a^{3} b^{2}$	+	$0 a^{2}$	+	$a b$	-	$4 a$
$3 a b$	+	$0$		$6 a^{4} b^{3}$	+	$0 a^{3}$	+	$3 a^{2} b^{2}$	-	$12 a^{2} b$	+	$0$
			-	$6 a^{4} b^{3}$	-	$0$
						$0$	+	$3 a^{2} b^{2}$	-	$12 a^{2} b$
							-	$3 a^{2} b^{2}$	-	$0$
									-	$12 a^{2} b$	+	$0$
									+	$12 a^{2} b$	-	$0$
												$0$

Step 16

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

$2 a^{3} b^{2} + 0 a^{2} + a b - 4 a$