Algebra Examples

$(- 20 m^{9} - 4 m^{8} - 14 m^{7}) \div 2 m^{5}$

Step 1

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

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

Step 2

Divide the highest order term in the dividend $- 20 m^{9}$ by the highest order term in divisor $2 m^{5}$ .

$10 m^{4}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

Step 3

Multiply the new quotient term by the divisor.

$10 m^{4}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 m^{9} + 0 + 0 + 0 + 0 + 0$

$10 m^{4}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$0$

Step 5

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

$10 m^{4}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$0$

$4 m^{8}$

$14 m^{7}$

$0$

Step 6

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

$10 m^{4}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$0$

$4 m^{8}$

$14 m^{7}$

$0$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

Step 7

Divide the highest order term in the dividend $- 4 m^{8}$ by the highest order term in divisor $2 m^{5}$ .

$10 m^{4}$

$2 m^{3}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$0$

$4 m^{8}$

$14 m^{7}$

$0$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

Step 8

Multiply the new quotient term by the divisor.

$10 m^{4}$

$2 m^{3}$

$2 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$4 m^{8}$

$14 m^{7}$

$0 m^{6}$

$0 m^{5}$

$0 m^{4}$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$20 m^{9}$

$0$

$4 m^{8}$

$14 m^{7}$

$0$

$0 m^{3}$

$0 m^{2}$

$0 m$

$0$

$4 m^{8}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 m^{8} + 0 + 0 + 0 + 0 + 0$

											-	$10 m^{4}$	-	$2 m^{3}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$

Step 10

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

											-	$10 m^{4}$	-	$2 m^{3}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$

Step 11

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

											-	$10 m^{4}$	-	$2 m^{3}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0 m^{2}$	+	$0 m$	+	$0$

Step 12

Divide the highest order term in the dividend $- 14 m^{7}$ by the highest order term in divisor $2 m^{5}$ .

											-	$10 m^{4}$	-	$2 m^{3}$	-	$7 m^{2}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0 m^{2}$	+	$0 m$	+	$0$

Step 13

Multiply the new quotient term by the divisor.

											-	$10 m^{4}$	-	$2 m^{3}$	-	$7 m^{2}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0 m^{2}$	+	$0 m$	+	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 m^{7} + 0 + 0 + 0 + 0 + 0$

											-	$10 m^{4}$	-	$2 m^{3}$	-	$7 m^{2}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0 m^{2}$	+	$0 m$	+	$0$
															+	$14 m^{7}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$

Step 15

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

											-	$10 m^{4}$	-	$2 m^{3}$	-	$7 m^{2}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0 m^{2}$	+	$0 m$	+	$0$
															+	$14 m^{7}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																										$0$

Step 16

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

											-	$10 m^{4}$	-	$2 m^{3}$	-	$7 m^{2}$
$2 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$	-	$20 m^{9}$	-	$4 m^{8}$	-	$14 m^{7}$	+	$0 m^{6}$	+	$0 m^{5}$	+	$0 m^{4}$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
											+	$20 m^{9}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
													-	$4 m^{8}$	-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0 m^{3}$	+	$0 m^{2}$	+	$0 m$	+	$0$
													+	$4 m^{8}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
															-	$14 m^{7}$	+	$0$	+	$0$	+	$0$	+	$0$	+	$0 m^{2}$	+	$0 m$	+	$0$
															+	$14 m^{7}$	-	$0$	-	$0$	-	$0$	-	$0$	-	$0$
																										$0$	+	$0 m$	+	$0$

Step 17

Since the remander is $0$ , the final answer is the quotient.

$- 10 m^{4} - 2 m^{3} - 7 m^{2}$