Algebra Examples

$(20 a^{4} b^{2} + 36 a^{3} b^{3} - 28 a^{2} b^{2}) \div 4 a^{2} b$

Step 1

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

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

Step 2

Divide the highest order term in the dividend $20 a^{4} b^{2}$ by the highest order term in divisor $4 a^{2} b$ .

$5 a^{2} b$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

Step 3

Multiply the new quotient term by the divisor.

$5 a^{2} b$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

$20 a^{4} b^{2}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $20 a^{4} b^{2} + 0 + 0$

$5 a^{2} b$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

$20 a^{4} b^{2}$

$0$

Step 5

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

$5 a^{2} b$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

$20 a^{4} b^{2}$

$0$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

Step 6

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

$5 a^{2} b$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

$20 a^{4} b^{2}$

$0$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

Step 7

Divide the highest order term in the dividend $36 a^{3} b^{3}$ by the highest order term in divisor $4 a^{2} b$ .

$5 a^{2} b$

$9 a b^{2}$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

$20 a^{4} b^{2}$

$0$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

Step 8

Multiply the new quotient term by the divisor.

$5 a^{2} b$

$9 a b^{2}$

$4 a^{2} b$

$0 a$

$0$

$20 a^{4} b^{2}$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$0$

$20 a^{4} b^{2}$

$0$

$36 a^{3} b^{3}$

$28 a^{2} b^{2}$

$0 a$

$36 a^{3} b^{3}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $36 a^{3} b^{3} + 0 + 0$

						$5 a^{2} b$	+	$9 a b^{2}$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$

Step 10

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

						$5 a^{2} b$	+	$9 a b^{2}$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$
									-	$28 a^{2} b^{2}$	+	$0$

Step 11

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

						$5 a^{2} b$	+	$9 a b^{2}$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$
									-	$28 a^{2} b^{2}$	+	$0$	+	$0$

Step 12

Divide the highest order term in the dividend $- 28 a^{2} b^{2}$ by the highest order term in divisor $4 a^{2} b$ .

						$5 a^{2} b$	+	$9 a b^{2}$	-	$7 b$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$
									-	$28 a^{2} b^{2}$	+	$0$	+	$0$

Step 13

Multiply the new quotient term by the divisor.

						$5 a^{2} b$	+	$9 a b^{2}$	-	$7 b$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$
									-	$28 a^{2} b^{2}$	+	$0$	+	$0$
									-	$28 b^{2} a^{2}$	+	$0$	+	$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 28 b^{2} a^{2} + 0 + 0$

						$5 a^{2} b$	+	$9 a b^{2}$	-	$7 b$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$
									-	$28 a^{2} b^{2}$	+	$0$	+	$0$
									+	$28 b^{2} a^{2}$	-	$0$	-	$0$

Step 15

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

						$5 a^{2} b$	+	$9 a b^{2}$	-	$7 b$
$4 a^{2} b$	+	$0 a$	+	$0$		$20 a^{4} b^{2}$	+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$	+	$0$
					-	$20 a^{4} b^{2}$	-	$0$	-	$0$
							+	$36 a^{3} b^{3}$	-	$28 a^{2} b^{2}$	+	$0 a$
							-	$36 a^{3} b^{3}$	-	$0$	-	$0$
									-	$28 a^{2} b^{2}$	+	$0$	+	$0$
									+	$28 b^{2} a^{2}$	-	$0$	-	$0$
														$0$

Step 16

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

$5 a^{2} b + 9 a b^{2} - 7 b$