Algebra Examples

$\frac{6 k^{2} m - 12 k^{3} m^{2} + 9 m^{3}}{2 (k m^{2})}$

Step 1

Expand $6 k^{2} m - 12 k^{3} m^{2} + 9 m^{3}$ .

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Step 1.1

Move $- 12 k^{3} m^{2}$ .

$\frac{6 k^{2} m + 9 m^{3} - 12 k^{3} m^{2}}{2 (k m^{2})}$

Step 1.2

Reorder $6 k^{2} m$ and $9 m^{3}$ .

$\frac{9 m^{3} + 6 k^{2} m - 12 k^{3} m^{2}}{2 (k m^{2})}$

Step 2

Remove parentheses.

$\frac{9 m^{3} + 6 k^{2} m - 12 k^{3} m^{2}}{2 k m^{2}}$

Step 3

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

$2 m^{2} k$

$0$

$12 k^{3} m^{2}$

$6 k^{2} m$

$0 k$

$9 m^{3}$

Step 4

Divide the highest order term in the dividend $- 12 k^{3} m^{2}$ by the highest order term in divisor $2 m^{2} k$ .

				-	$6 k^{2}$
	$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$

Step 5

Multiply the new quotient term by the divisor.

			-	$6 k^{2}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			-	$12 k^{3} m^{2}$	+	$0$

Step 6

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 k^{3} m^{2} + 0$

			-	$6 k^{2}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$

Step 7

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

			-	$6 k^{2}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$

Step 8

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

			-	$6 k^{2}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$	+	$0 k$

Step 9

Divide the highest order term in the dividend $6 k^{2} m$ by the highest order term in divisor $2 m^{2} k$ .

			-	$6 k^{2}$	+	$\frac{3 k}{m}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$	+	$0 k$

Step 10

Multiply the new quotient term by the divisor.

			-	$6 k^{2}$	+	$\frac{3 k}{m}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$	+	$0 k$
					+	$6 k^{2} m$	+	$0$

Step 11

The expression needs to be subtracted from the dividend, so change all the signs in $6 k^{2} m + 0$

			-	$6 k^{2}$	+	$\frac{3 k}{m}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$	+	$0 k$
					-	$6 k^{2} m$	-	$0$

Step 12

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

			-	$6 k^{2}$	+	$\frac{3 k}{m}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$	+	$0 k$
					-	$6 k^{2} m$	-	$0$
								$0$

Step 13

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

			-	$6 k^{2}$	+	$\frac{3 k}{m}$
$2 m^{2} k$	+	$0$	-	$12 k^{3} m^{2}$	+	$6 k^{2} m$	+	$0 k$	+	$9 m^{3}$
			+	$12 k^{3} m^{2}$	-	$0$
					+	$6 k^{2} m$	+	$0 k$
					-	$6 k^{2} m$	-	$0$
								$0$	+	$9 m^{3}$

Step 14

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

$- 6 k^{2} + \frac{3 k}{m} + \frac{9 m^{3}}{2 m^{2} k}$