Examples

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

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

x + 1

$x + 1$

Step 1

Divide the higher order polynomial by the other polynomial in order to find the remainder.

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

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

Step 2

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

0

$0$ .

x

$x$

1

$1$

2 x^{3}

$2 x^{3}$

3 x^{2}

$3 x^{2}$

4 x

$4 x$

1

$1$

Step 3

Divide the highest order term in the dividend

2 x^{3}

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

x

$x$ .

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

Step 4

Multiply the new quotient term by the divisor.

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

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in

2 x^{3} + 2 x^{2}

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

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

Step 6

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

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

Step 7

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

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

Step 8

Divide the highest order term in the dividend

$- 5 x^{2}$ by the highest order term in divisor

$x$ .

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

Step 9

Multiply the new quotient term by the divisor.

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

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in

$- 5 x^{2} - 5 x$

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

Step 11

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

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

Step 12

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

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

Step 13

Divide the highest order term in the dividend

$9 x$ by the highest order term in divisor

$x$ .

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

Step 14

Multiply the new quotient term by the divisor.

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

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in

$9 x + 9$

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

Step 16

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

				$2 x^{2}$	-	$5 x$	+	$9$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	+	$4 x$	-	$1$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	+	$4 x$
					+	$5 x^{2}$	+	$5 x$
							+	$9 x$	-	$1$
							-	$9 x$	-	$9$
									-	$10$

Step 17

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

$2 x^{2} - 5 x + 9 - \frac{10}{x + 1}$

Step 18

The remainder is the part of the answer that is left after the division by

$x + 1$ is complete.

$- 10$

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