Algebra Examples

$\frac{90 x^{5} y^{6} - 108 x^{2} y^{3} - 126 x}{- 9 x}$

Step 1

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

$9 x$

$0$

$90 y^{6} x^{5}$

$0 x^{4}$

$0 x^{3}$

$108 y^{3} x^{2}$

$126 x$

$0$

Step 2

Divide the highest order term in the dividend $90 y^{6} x^{5}$ by the highest order term in divisor $- 9 x$ .

$10 y^{6} x^{4}$

$9 x$

$0$

$90 y^{6} x^{5}$

$0 x^{4}$

$0 x^{3}$

$108 y^{3} x^{2}$

$126 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$10 y^{6} x^{4}$

$9 x$

$0$

$90 y^{6} x^{5}$

$0 x^{4}$

$0 x^{3}$

$108 y^{3} x^{2}$

$126 x$

$0$

$90 y^{6} x^{5}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $90 y^{6} x^{5} + 0$

				-	$10 y^{6} x^{4}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$

Step 5

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

				-	$10 y^{6} x^{4}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$

Step 6

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

				-	$10 y^{6} x^{4}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$

Step 7

Divide the highest order term in the dividend $0 x^{3}$ by the highest order term in divisor $- 9 x$ .

$10 y^{6} x^{4}$

$0 x^{3}$

$9 x$

$0$

$90 y^{6} x^{5}$

$0 x^{4}$

$0 x^{3}$

$108 y^{3} x^{2}$

$126 x$

$0$

$90 y^{6} x^{5}$

$0$

$0 x^{3}$

$108 y^{3} x^{2}$

Step 8

Multiply the new quotient term by the divisor.

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								+	$0$	+	$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $0 + 0$

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$

Step 10

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

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$

Step 11

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

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$

Step 12

Divide the highest order term in the dividend $- 108 y^{3} x^{2}$ by the highest order term in divisor $- 9 x$ .

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$

Step 13

Multiply the new quotient term by the divisor.

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										-	$108 y^{3} x^{2}$	+	$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 108 y^{3} x^{2} + 0$

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$

Step 15

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

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$
												-	$126 x$

Step 16

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

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$
												-	$126 x$	+	$0$

Step 17

Divide the highest order term in the dividend $- 126 x$ by the highest order term in divisor $- 9 x$ .

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$	+	$14$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$
												-	$126 x$	+	$0$

Step 18

Multiply the new quotient term by the divisor.

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$	+	$14$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$
												-	$126 x$	+	$0$
												-	$126 x$	+	$0$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 126 x + 0$

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$	+	$14$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$
												-	$126 x$	+	$0$
												+	$126 x$	-	$0$

Step 20

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

				-	$10 y^{6} x^{4}$	+	$0 x^{3}$	+	$12 y^{3} x$	+	$14$
-	$9 x$	+	$0$		$90 y^{6} x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$	-	$126 x$	+	$0$
				-	$90 y^{6} x^{5}$	-	$0$
							$0$	+	$0 x^{3}$	-	$108 y^{3} x^{2}$
								-	$0$	-	$0$
										-	$108 y^{3} x^{2}$	-	$126 x$
										+	$108 y^{3} x^{2}$	-	$0$
												-	$126 x$	+	$0$
												+	$126 x$	-	$0$
															$0$

Step 21

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

$- 10 y^{6} x^{4} + 0 x^{3} + 12 y^{3} x + 14$