Algebra Examples

$\frac{- 2 m^{3} - m + m^{2} + 1}{m + 1}$

Step 1

Move $- m$ .

$\frac{- 2 m^{3} + m^{2} - m + 1}{m + 1}$

Step 2

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

$m$

$1$

$2 m^{3}$

$m^{2}$

$m$

$1$

Step 3

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

				-	$2 m^{2}$
	$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$

Step 4

Multiply the new quotient term by the divisor.

			-	$2 m^{2}$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			-	$2 m^{3}$	-	$2 m^{2}$

Step 5

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

			-	$2 m^{2}$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			+	$2 m^{3}$	+	$2 m^{2}$

Step 6

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

			-	$2 m^{2}$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			+	$2 m^{3}$	+	$2 m^{2}$
					+	$3 m^{2}$

Step 7

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

			-	$2 m^{2}$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			+	$2 m^{3}$	+	$2 m^{2}$
					+	$3 m^{2}$	-	$m$

Step 8

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

			-	$2 m^{2}$	+	$3 m$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			+	$2 m^{3}$	+	$2 m^{2}$
					+	$3 m^{2}$	-	$m$

Step 9

Multiply the new quotient term by the divisor.

			-	$2 m^{2}$	+	$3 m$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			+	$2 m^{3}$	+	$2 m^{2}$
					+	$3 m^{2}$	-	$m$
					+	$3 m^{2}$	+	$3 m$

Step 10

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

			-	$2 m^{2}$	+	$3 m$
$m$	+	$1$	-	$2 m^{3}$	+	$m^{2}$	-	$m$	+	$1$
			+	$2 m^{3}$	+	$2 m^{2}$
					+	$3 m^{2}$	-	$m$
					-	$3 m^{2}$	-	$3 m$

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Multiply the new quotient term by the divisor.

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

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 m - 4$

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

Step 16

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

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

Step 17

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

$- 2 m^{2} + 3 m - 4 + \frac{5}{m + 1}$