Algebra Examples

$(18 x^{6} y^{4} + 24 x^{5} y^{3} - 30 x^{3} y^{2}) \div (6 x^{3} y^{2})$

Step 1

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

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $18 x^{6} y^{4}$ by the highest order term in divisor $6 x^{3} y^{2}$ .

$3 x^{3} y^{2}$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{3} y^{2}$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{6} y^{4} + 0 + 0 + 0$

$3 x^{3} y^{2}$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

Step 5

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

$3 x^{3} y^{2}$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

Step 6

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

$3 x^{3} y^{2}$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $24 x^{5} y^{3}$ by the highest order term in divisor $6 x^{3} y^{2}$ .

$3 x^{3} y^{2}$

$4 x^{2} y$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{3} y^{2}$

$4 x^{2} y$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$24 x^{5} y^{3}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $24 x^{5} y^{3} + 0 + 0 + 0$

$3 x^{3} y^{2}$

$4 x^{2} y$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$24 x^{5} y^{3}$

$0$

Step 10

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

$3 x^{3} y^{2}$

$4 x^{2} y$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0$

Step 11

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

$3 x^{3} y^{2}$

$4 x^{2} y$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0$

$0 x$

$0$

Step 12

Divide the highest order term in the dividend $- 30 x^{3} y^{2}$ by the highest order term in divisor $6 x^{3} y^{2}$ .

$3 x^{3} y^{2}$

$4 x^{2} y$

$0 x$

$5$

$6 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$24 x^{5} y^{3}$

$0 x^{4}$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$0$

$18 x^{6} y^{4}$

$0$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0 x^{2}$

$0 x$

$24 x^{5} y^{3}$

$0$

$30 x^{3} y^{2}$

$0$

$0 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

								$3 x^{3} y^{2}$	+	$4 x^{2} y$	+	$0 x$	-	$5$
$6 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$	+	$0$		$18 x^{6} y^{4}$	+	$24 x^{5} y^{3}$	+	$0 x^{4}$	-	$30 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$	+	$0$
							-	$18 x^{6} y^{4}$	-	$0$	-	$0$	-	$0$
									+	$24 x^{5} y^{3}$	+	$0$	-	$30 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$
									-	$24 x^{5} y^{3}$	-	$0$	-	$0$	-	$0$
													-	$30 x^{3} y^{2}$	+	$0$	+	$0 x$	+	$0$
													-	$30 x^{3} y^{2}$	+	$0$	+	$0$	+	$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 x^{3} y^{2} + 0 + 0 + 0$

								$3 x^{3} y^{2}$	+	$4 x^{2} y$	+	$0 x$	-	$5$
$6 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$	+	$0$		$18 x^{6} y^{4}$	+	$24 x^{5} y^{3}$	+	$0 x^{4}$	-	$30 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$	+	$0$
							-	$18 x^{6} y^{4}$	-	$0$	-	$0$	-	$0$
									+	$24 x^{5} y^{3}$	+	$0$	-	$30 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$
									-	$24 x^{5} y^{3}$	-	$0$	-	$0$	-	$0$
													-	$30 x^{3} y^{2}$	+	$0$	+	$0 x$	+	$0$
													+	$30 x^{3} y^{2}$	-	$0$	-	$0$	-	$0$

Step 15

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

								$3 x^{3} y^{2}$	+	$4 x^{2} y$	+	$0 x$	-	$5$
$6 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$	+	$0$		$18 x^{6} y^{4}$	+	$24 x^{5} y^{3}$	+	$0 x^{4}$	-	$30 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$	+	$0$
							-	$18 x^{6} y^{4}$	-	$0$	-	$0$	-	$0$
									+	$24 x^{5} y^{3}$	+	$0$	-	$30 x^{3} y^{2}$	+	$0 x^{2}$	+	$0 x$
									-	$24 x^{5} y^{3}$	-	$0$	-	$0$	-	$0$
													-	$30 x^{3} y^{2}$	+	$0$	+	$0 x$	+	$0$
													+	$30 x^{3} y^{2}$	-	$0$	-	$0$	-	$0$
																				$0$

Step 16

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

$3 x^{3} y^{2} + 4 x^{2} y + 0 x - 5$