Precalculus Examples

$\frac{m^{4} - 6 m^{3} + 17 m^{2} - 28 m + 20}{m^{2} + 2 m + 5}$

Step 1

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

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

Step 2

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

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

Step 3

Multiply the new quotient term by the divisor.

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $m^{4} + 2 m^{3} + 5 m^{2}$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

Step 5

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

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

Step 6

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

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

Step 7

Divide the highest order term in the dividend $- 8 m^{3}$ by the highest order term in divisor $m^{2}$ .

$m^{2}$

$8 m$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

Step 8

Multiply the new quotient term by the divisor.

$m^{2}$

$8 m$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 m^{3} - 16 m^{2} - 40 m$

$m^{2}$

$8 m$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

Step 10

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

$m^{2}$

$8 m$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

$28 m^{2}$

$12 m$

Step 11

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

$m^{2}$

$8 m$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

$28 m^{2}$

$12 m$

$20$

Step 12

Divide the highest order term in the dividend $28 m^{2}$ by the highest order term in divisor $m^{2}$ .

$m^{2}$

$8 m$

$28$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

$28 m^{2}$

$12 m$

$20$

Step 13

Multiply the new quotient term by the divisor.

$m^{2}$

$8 m$

$28$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

$28 m^{2}$

$12 m$

$20$

$28 m^{2}$

$56 m$

$140$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $28 m^{2} + 56 m + 140$

$m^{2}$

$8 m$

$28$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

$28 m^{2}$

$12 m$

$20$

$28 m^{2}$

$56 m$

$140$

Step 15

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

$m^{2}$

$8 m$

$28$

$m^{2}$

$2 m$

$5$

$m^{4}$

$6 m^{3}$

$17 m^{2}$

$28 m$

$20$

$m^{4}$

$2 m^{3}$

$5 m^{2}$

$8 m^{3}$

$12 m^{2}$

$28 m$

$8 m^{3}$

$16 m^{2}$

$40 m$

$28 m^{2}$

$12 m$

$20$

$28 m^{2}$

$56 m$

$140$

$44 m$

$120$

Step 16

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

$m^{2} - 8 m + 28 + \frac{- 44 m - 120}{m^{2} + 2 m + 5}$