Algebra Examples

$(- 30 x^{3} y + 12 x^{2} y^{2} - 18 x^{2} y) \div (- 6 x^{2} y)$

Step 1

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

$6 x^{2} y$

$0 x$

$0$

$30 x^{3} y$

$12 x^{2} y^{2}$

$18 x^{2} y$

$0$

Step 2

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

$5 x$

$6 x^{2} y$

$0 x$

$0$

$30 x^{3} y$

$12 x^{2} y^{2}$

$18 x^{2} y$

$0$

Step 3

Multiply the new quotient term by the divisor.

$5 x$

$6 x^{2} y$

$0 x$

$0$

$30 x^{3} y$

$12 x^{2} y^{2}$

$18 x^{2} y$

$0$

$30 x^{3} y$

$0$

Step 4

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

							$5 x$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$

Step 5

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

							$5 x$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$

Step 6

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

							$5 x$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$

Step 7

Divide the highest order term in the dividend $12 x^{2} y^{2}$ by the highest order term in divisor $- 6 x^{2} y$ .

							$5 x$	-	$2 y$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$

Step 8

Multiply the new quotient term by the divisor.

							$5 x$	-	$2 y$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								+	$12 y^{2} x^{2}$	+	$0$	+	$0$

Step 9

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

							$5 x$	-	$2 y$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								-	$12 y^{2} x^{2}$	-	$0$	-	$0$

Step 10

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

							$5 x$	-	$2 y$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								-	$12 y^{2} x^{2}$	-	$0$	-	$0$
										-	$18 x^{2} y$	+	$0$

Step 11

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

							$5 x$	-	$2 y$	+	$3$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								-	$12 y^{2} x^{2}$	-	$0$	-	$0$
										-	$18 x^{2} y$	+	$0$

Step 12

Multiply the new quotient term by the divisor.

							$5 x$	-	$2 y$	+	$3$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								-	$12 y^{2} x^{2}$	-	$0$	-	$0$
										-	$18 x^{2} y$	+	$0$
										-	$18 x^{2} y$	+	$0$	+	$0$

Step 13

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{2} y + 0 + 0$

							$5 x$	-	$2 y$	+	$3$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								-	$12 y^{2} x^{2}$	-	$0$	-	$0$
										-	$18 x^{2} y$	+	$0$
										+	$18 x^{2} y$	-	$0$	-	$0$

Step 14

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

							$5 x$	-	$2 y$	+	$3$
-	$6 x^{2} y$	+	$0 x$	+	$0$	-	$30 x^{3} y$	+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
						+	$30 x^{3} y$	-	$0$	-	$0$
								+	$12 x^{2} y^{2}$	-	$18 x^{2} y$	+	$0$
								-	$12 y^{2} x^{2}$	-	$0$	-	$0$
										-	$18 x^{2} y$	+	$0$
										+	$18 x^{2} y$	-	$0$	-	$0$
															$0$

Step 15

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

$5 x - 2 y + 3$