Divide Using Long Polynomial Division (6x^3+x-1)÷(x+2)

$(6 x^{3} + x - 1) \div (x + 2)$

Step 1

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

$x$

$2$

$6 x^{3}$

$0 x^{2}$

$x$

$1$

Step 2

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

					$6 x^{2}$
	$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$

Step 3

Multiply the new quotient term by the divisor.

				$6 x^{2}$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			+	$6 x^{3}$	+	$12 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 12 x^{2}$

				$6 x^{2}$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$

Step 5

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

				$6 x^{2}$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$

Step 6

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

				$6 x^{2}$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$

Step 7

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

				$6 x^{2}$	-	$12 x$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$

Step 8

Multiply the new quotient term by the divisor.

				$6 x^{2}$	-	$12 x$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					-	$12 x^{2}$	-	$24 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} - 24 x$

				$6 x^{2}$	-	$12 x$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$

Step 10

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

				$6 x^{2}$	-	$12 x$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$
							+	$25 x$

Step 11

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

				$6 x^{2}$	-	$12 x$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$
							+	$25 x$	-	$1$

Step 12

Divide the highest order term in the dividend $25 x$ by the highest order term in divisor $x$ .

				$6 x^{2}$	-	$12 x$	+	$25$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$
							+	$25 x$	-	$1$

Step 13

Multiply the new quotient term by the divisor.

				$6 x^{2}$	-	$12 x$	+	$25$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$
							+	$25 x$	-	$1$
							+	$25 x$	+	$50$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x + 50$

				$6 x^{2}$	-	$12 x$	+	$25$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$
							+	$25 x$	-	$1$
							-	$25 x$	-	$50$

Step 15

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

				$6 x^{2}$	-	$12 x$	+	$25$
$x$	+	$2$		$6 x^{3}$	+	$0 x^{2}$	+	$x$	-	$1$
			-	$6 x^{3}$	-	$12 x^{2}$
					-	$12 x^{2}$	+	$x$
					+	$12 x^{2}$	+	$24 x$
							+	$25 x$	-	$1$
							-	$25 x$	-	$50$
									-	$51$

Step 16

The final answer is the quotient plus the remainder over the divisor.

$6 x^{2} - 12 x + 25 - \frac{51}{x + 2}$

Divide Using Long Polynomial Division (6x^3+x-1)÷(x+2)

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