Divide (-18x^3+17x+6)÷(3x+2)

$(- 18 x^{3} + 17 x + 6) \div (3 x + 2)$

Step 1

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

$3 x$

$2$

$18 x^{3}$

$0 x^{2}$

$17 x$

$6$

Step 2

Divide the highest order term in the dividend $- 18 x^{3}$ by the highest order term in divisor $3 x$ .

				-	$6 x^{2}$
	$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$

Step 3

Multiply the new quotient term by the divisor.

			-	$6 x^{2}$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			-	$18 x^{3}$	-	$12 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{3} - 12 x^{2}$

			-	$6 x^{2}$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$

Step 5

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

			-	$6 x^{2}$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$

Step 6

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

			-	$6 x^{2}$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$

Step 7

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

			-	$6 x^{2}$	+	$4 x$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$

Step 8

Multiply the new quotient term by the divisor.

			-	$6 x^{2}$	+	$4 x$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					+	$12 x^{2}$	+	$8 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} + 8 x$

			-	$6 x^{2}$	+	$4 x$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$

Step 10

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

			-	$6 x^{2}$	+	$4 x$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$
							+	$9 x$

Step 11

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

			-	$6 x^{2}$	+	$4 x$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$
							+	$9 x$	+	$6$

Step 12

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $3 x$ .

			-	$6 x^{2}$	+	$4 x$	+	$3$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$
							+	$9 x$	+	$6$

Step 13

Multiply the new quotient term by the divisor.

			-	$6 x^{2}$	+	$4 x$	+	$3$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$
							+	$9 x$	+	$6$
							+	$9 x$	+	$6$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $9 x + 6$

			-	$6 x^{2}$	+	$4 x$	+	$3$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$
							+	$9 x$	+	$6$
							-	$9 x$	-	$6$

Step 15

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

			-	$6 x^{2}$	+	$4 x$	+	$3$
$3 x$	+	$2$	-	$18 x^{3}$	+	$0 x^{2}$	+	$17 x$	+	$6$
			+	$18 x^{3}$	+	$12 x^{2}$
					+	$12 x^{2}$	+	$17 x$
					-	$12 x^{2}$	-	$8 x$
							+	$9 x$	+	$6$
							-	$9 x$	-	$6$
										$0$

Step 16

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

$- 6 x^{2} + 4 x + 3$

Divide (-18x^3+17x+6)÷(3x+2)

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