Divide Using Long Polynomial Division (6x^5+2x^4-18x^3-2x^2+12x)/(2x^3-2x)

$\frac{6 x^{5} + 2 x^{4} - 18 x^{3} - 2 x^{2} + 12 x}{2 x^{3} - 2 x}$

Step 1

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

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

Step 2

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

$3 x^{2}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{2}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{5} + 0 - 6 x^{3} + 0$

$3 x^{2}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

Step 5

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

$3 x^{2}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

Step 6

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

$3 x^{2}$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

Step 7

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

$3 x^{2}$

$x$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{2}$

$x$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

$2 x^{4}$

$0$

$2 x^{2}$

$0$

Step 9

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

$3 x^{2}$

$x$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

$2 x^{4}$

$0$

$2 x^{2}$

$0$

Step 10

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

$3 x^{2}$

$x$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

$2 x^{4}$

$0$

$2 x^{2}$

$0$

$12 x^{3}$

$0$

$12 x$

Step 11

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

$3 x^{2}$

$x$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

$2 x^{4}$

$0$

$2 x^{2}$

$0$

$12 x^{3}$

$0$

$12 x$

$0$

Step 12

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

								$3 x^{2}$	+	$x$	-	$6$
$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$0$		$6 x^{5}$	+	$2 x^{4}$	-	$18 x^{3}$	-	$2 x^{2}$	+	$12 x$	+	$0$
							-	$6 x^{5}$	-	$0$	+	$6 x^{3}$	-	$0$
									+	$2 x^{4}$	-	$12 x^{3}$	-	$2 x^{2}$	+	$12 x$
									-	$2 x^{4}$	-	$0$	+	$2 x^{2}$	-	$0$
											-	$12 x^{3}$	+	$0$	+	$12 x$	+	$0$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{2}$

$x$

$6$

$2 x^{3}$

$0 x^{2}$

$2 x$

$0$

$6 x^{5}$

$2 x^{4}$

$18 x^{3}$

$2 x^{2}$

$12 x$

$0$

$6 x^{5}$

$0$

$6 x^{3}$

$0$

$2 x^{4}$

$12 x^{3}$

$2 x^{2}$

$12 x$

$2 x^{4}$

$0$

$2 x^{2}$

$0$

$12 x^{3}$

$0$

$12 x$

$0$

$12 x^{3}$

$0$

$12 x$

$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{3} + 0 + 12 x + 0$

								$3 x^{2}$	+	$x$	-	$6$
$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$0$		$6 x^{5}$	+	$2 x^{4}$	-	$18 x^{3}$	-	$2 x^{2}$	+	$12 x$	+	$0$
							-	$6 x^{5}$	-	$0$	+	$6 x^{3}$	-	$0$
									+	$2 x^{4}$	-	$12 x^{3}$	-	$2 x^{2}$	+	$12 x$
									-	$2 x^{4}$	-	$0$	+	$2 x^{2}$	-	$0$
											-	$12 x^{3}$	+	$0$	+	$12 x$	+	$0$
											+	$12 x^{3}$	-	$0$	-	$12 x$	-	$0$

Step 15

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

								$3 x^{2}$	+	$x$	-	$6$
$2 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$0$		$6 x^{5}$	+	$2 x^{4}$	-	$18 x^{3}$	-	$2 x^{2}$	+	$12 x$	+	$0$
							-	$6 x^{5}$	-	$0$	+	$6 x^{3}$	-	$0$
									+	$2 x^{4}$	-	$12 x^{3}$	-	$2 x^{2}$	+	$12 x$
									-	$2 x^{4}$	-	$0$	+	$2 x^{2}$	-	$0$
											-	$12 x^{3}$	+	$0$	+	$12 x$	+	$0$
											+	$12 x^{3}$	-	$0$	-	$12 x$	-	$0$
																		$0$

Step 16

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

$3 x^{2} + x - 6$

Divide Using Long Polynomial Division (6x^5+2x^4-18x^3-2x^2+12x)/(2x^3-2x)

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