Algebra Examples

$(s^{4} + 5 s^{3} - 12 s^{2} + 13 s - 2) \div (s - 1)$

Step 1

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

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

Step 2

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

$s^{3}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

Step 3

Multiply the new quotient term by the divisor.

$s^{3}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $s^{4} - s^{3}$

$s^{3}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

Step 5

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

$s^{3}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

Step 6

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

$s^{3}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

Step 7

Divide the highest order term in the dividend $6 s^{3}$ by the highest order term in divisor $s$ .

$s^{3}$

$6 s^{2}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

Step 8

Multiply the new quotient term by the divisor.

$s^{3}$

$6 s^{2}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $6 s^{3} - 6 s^{2}$

$s^{3}$

$6 s^{2}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

Step 10

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

$s^{3}$

$6 s^{2}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

Step 11

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

$s^{3}$

$6 s^{2}$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

Step 12

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

$s^{3}$

$6 s^{2}$

$6 s$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

Step 13

Multiply the new quotient term by the divisor.

$s^{3}$

$6 s^{2}$

$6 s$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

Step 14

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

$s^{3}$

$6 s^{2}$

$6 s$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

Step 15

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

$s^{3}$

$6 s^{2}$

$6 s$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

$7 s$

Step 16

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

$s^{3}$

$6 s^{2}$

$6 s$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

$7 s$

$2$

Step 17

Divide the highest order term in the dividend $7 s$ by the highest order term in divisor $s$ .

$s^{3}$

$6 s^{2}$

$6 s$

$7$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

$7 s$

$2$

Step 18

Multiply the new quotient term by the divisor.

$s^{3}$

$6 s^{2}$

$6 s$

$7$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

$7 s$

$2$

$7 s$

$7$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $7 s - 7$

$s^{3}$

$6 s^{2}$

$6 s$

$7$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

$7 s$

$2$

$7 s$

$7$

Step 20

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

$s^{3}$

$6 s^{2}$

$6 s$

$7$

$s$

$1$

$s^{4}$

$5 s^{3}$

$12 s^{2}$

$13 s$

$2$

$s^{4}$

$s^{3}$

$6 s^{3}$

$12 s^{2}$

$6 s^{3}$

$6 s^{2}$

$13 s$

$6 s^{2}$

$6 s$

$7 s$

$2$

$7 s$

$7$

$5$

Step 21

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

$s^{3} + 6 s^{2} - 6 s + 7 + \frac{5}{s - 1}$