Algebra Examples

$\frac{a^{4} - a}{a + 1}$

Step 1

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

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

Step 2

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

$a^{3}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

Step 3

Multiply the new quotient term by the divisor.

$a^{3}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

Step 4

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

$a^{3}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

Step 5

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

$a^{3}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

Step 6

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

$a^{3}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

Step 7

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

Step 8

Multiply the new quotient term by the divisor.

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 9

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 10

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

Step 11

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

Step 12

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

Step 13

Multiply the new quotient term by the divisor.

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

Step 14

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

Step 15

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$2 a$

Step 16

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

$a^{3}$

$a^{2}$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$2 a$

$0$

Step 17

Divide the highest order term in the dividend $- 2 a$ by the highest order term in divisor $a$ .

$a^{3}$

$a^{2}$

$a$

$2$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$2 a$

$0$

Step 18

Multiply the new quotient term by the divisor.

$a^{3}$

$a^{2}$

$a$

$2$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$2 a$

$0$

$2 a$

$2$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 a - 2$

$a^{3}$

$a^{2}$

$a$

$2$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$2 a$

$0$

$2 a$

$2$

Step 20

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

$a^{3}$

$a^{2}$

$a$

$2$

$a$

$1$

$a^{4}$

$0 a^{3}$

$0 a^{2}$

$a$

$0$

$a^{4}$

$a^{3}$

$0 a^{2}$

$a^{3}$

$a^{2}$

$a$

$a^{2}$

$a$

$2 a$

$0$

$2 a$

$2$

Step 21

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

$a^{3} - a^{2} + a - 2 + \frac{2}{a + 1}$