Divide Using Long Polynomial Division (9x^3-18x^2-x+2)÷(3x+1)

$(9 x^{3} - 18 x^{2} - x + 2) \div (3 x + 1)$

Step 1

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

$3 x$

$1$

$9 x^{3}$

$18 x^{2}$

$x$

$2$

Step 2

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

					$3 x^{2}$
	$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$

Step 3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			+	$9 x^{3}$	+	$3 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} + 3 x^{2}$

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$

Step 5

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$

Step 6

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$

Step 7

Divide the highest order term in the dividend $- 21 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$7 x$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$7 x$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					-	$21 x^{2}$	-	$7 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 21 x^{2} - 7 x$

				$3 x^{2}$	-	$7 x$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$

Step 10

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

				$3 x^{2}$	-	$7 x$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$
							+	$6 x$

Step 11

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

				$3 x^{2}$	-	$7 x$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$
							+	$6 x$	+	$2$

Step 12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$7 x$	+	$2$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$
							+	$6 x$	+	$2$

Step 13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$7 x$	+	$2$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$
							+	$6 x$	+	$2$
							+	$6 x$	+	$2$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 2$

				$3 x^{2}$	-	$7 x$	+	$2$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$
							+	$6 x$	+	$2$
							-	$6 x$	-	$2$

Step 15

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

				$3 x^{2}$	-	$7 x$	+	$2$
$3 x$	+	$1$		$9 x^{3}$	-	$18 x^{2}$	-	$x$	+	$2$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$21 x^{2}$	-	$x$
					+	$21 x^{2}$	+	$7 x$
							+	$6 x$	+	$2$
							-	$6 x$	-	$2$
										$0$

Step 16

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

$3 x^{2} - 7 x + 2$

Divide Using Long Polynomial Division (9x^3-18x^2-x+2)÷(3x+1)

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