Divide Using Long Polynomial Division (2x^3+5x^2-21x+1)/(x-2)

$\frac{2 x^{3} + 5 x^{2} - 21 x + 1}{x - 2}$

Step 1

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

$x$

$2$

$2 x^{3}$

$5 x^{2}$

$21 x$

$1$

Step 2

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

					$2 x^{2}$
	$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$

Step 3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 4 x^{2}$

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 5

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$

Step 6

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$

Step 7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$9 x$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$9 x$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					+	$9 x^{2}$	-	$18 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 18 x$

				$2 x^{2}$	+	$9 x$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$

Step 10

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

				$2 x^{2}$	+	$9 x$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$
							-	$3 x$

Step 11

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

				$2 x^{2}$	+	$9 x$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$
							-	$3 x$	+	$1$

Step 12

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

				$2 x^{2}$	+	$9 x$	-	$3$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$
							-	$3 x$	+	$1$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$9 x$	-	$3$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$
							-	$3 x$	+	$1$
							-	$3 x$	+	$6$

Step 14

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

				$2 x^{2}$	+	$9 x$	-	$3$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$
							-	$3 x$	+	$1$
							+	$3 x$	-	$6$

Step 15

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

				$2 x^{2}$	+	$9 x$	-	$3$
$x$	-	$2$		$2 x^{3}$	+	$5 x^{2}$	-	$21 x$	+	$1$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$9 x^{2}$	-	$21 x$
					-	$9 x^{2}$	+	$18 x$
							-	$3 x$	+	$1$
							+	$3 x$	-	$6$
									-	$5$

Step 16

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

$2 x^{2} + 9 x - 3 - \frac{5}{x - 2}$

Divide Using Long Polynomial Division (2x^3+5x^2-21x+1)/(x-2)

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