Algebra Examples

$(2 m^{4} + 21 m^{3} + 35 m^{2} - 37 m + 45) \div (2 m + 7)$

Step 1

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

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

Step 2

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

$m^{3}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

Step 3

Multiply the new quotient term by the divisor.

$m^{3}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 m^{4} + 7 m^{3}$

$m^{3}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

Step 5

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

$m^{3}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

Step 6

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

$m^{3}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

Step 7

Divide the highest order term in the dividend $14 m^{3}$ by the highest order term in divisor $2 m$ .

$m^{3}$

$7 m^{2}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

Step 8

Multiply the new quotient term by the divisor.

$m^{3}$

$7 m^{2}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $14 m^{3} + 49 m^{2}$

$m^{3}$

$7 m^{2}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

Step 10

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

$m^{3}$

$7 m^{2}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

Step 11

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

$m^{3}$

$7 m^{2}$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

Step 12

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

$m^{3}$

$7 m^{2}$

$7 m$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

Step 13

Multiply the new quotient term by the divisor.

$m^{3}$

$7 m^{2}$

$7 m$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 m^{2} - 49 m$

$m^{3}$

$7 m^{2}$

$7 m$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

Step 15

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

$m^{3}$

$7 m^{2}$

$7 m$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

$12 m$

Step 16

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

$m^{3}$

$7 m^{2}$

$7 m$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

$12 m$

$45$

Step 17

Divide the highest order term in the dividend $12 m$ by the highest order term in divisor $2 m$ .

$m^{3}$

$7 m^{2}$

$7 m$

$6$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

$12 m$

$45$

Step 18

Multiply the new quotient term by the divisor.

$m^{3}$

$7 m^{2}$

$7 m$

$6$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

$12 m$

$45$

$12 m$

$42$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $12 m + 42$

$m^{3}$

$7 m^{2}$

$7 m$

$6$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

$12 m$

$45$

$12 m$

$42$

Step 20

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

$m^{3}$

$7 m^{2}$

$7 m$

$6$

$2 m$

$7$

$2 m^{4}$

$21 m^{3}$

$35 m^{2}$

$37 m$

$45$

$2 m^{4}$

$7 m^{3}$

$14 m^{3}$

$35 m^{2}$

$14 m^{3}$

$49 m^{2}$

$14 m^{2}$

$37 m$

$14 m^{2}$

$49 m$

$12 m$

$45$

$12 m$

$42$

$3$

Step 21

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

$m^{3} + 7 m^{2} - 7 m + 6 + \frac{3}{2 m + 7}$