Divide Using Long Polynomial Division (2x^3+4x^2-2x-1)/(x+3)

$\frac{2 x^{3} + 4 x^{2} - 2 x - 1}{x + 3}$

Step 1

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

$x$

$3$

$2 x^{3}$

$4 x^{2}$

$2 x$

$1$

Step 2

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

					$2 x^{2}$
	$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$

Step 3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			+	$2 x^{3}$	+	$6 x^{2}$

Step 4

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

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$

Step 5

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

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$

Step 6

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

				$2 x^{2}$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$

Step 7

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					-	$2 x^{2}$	-	$6 x$

Step 9

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$

Step 10

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$
							+	$4 x$

Step 11

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

				$2 x^{2}$	-	$2 x$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$
							+	$4 x$	-	$1$

Step 12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$2 x$	+	$4$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$
							+	$4 x$	-	$1$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$2 x$	+	$4$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$
							+	$4 x$	-	$1$
							+	$4 x$	+	$12$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 12$

				$2 x^{2}$	-	$2 x$	+	$4$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$
							+	$4 x$	-	$1$
							-	$4 x$	-	$12$

Step 15

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

				$2 x^{2}$	-	$2 x$	+	$4$
$x$	+	$3$		$2 x^{3}$	+	$4 x^{2}$	-	$2 x$	-	$1$
			-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{2}$	-	$2 x$
					+	$2 x^{2}$	+	$6 x$
							+	$4 x$	-	$1$
							-	$4 x$	-	$12$
									-	$13$

Step 16

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

$2 x^{2} - 2 x + 4 - \frac{13}{x + 3}$

Divide Using Long Polynomial Division (2x^3+4x^2-2x-1)/(x+3)

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