Algebra Examples

$\frac{(r^{4} - r^{2} + 4)}{(r^{2} - r + 2)}$

Step 1

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

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

Step 2

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

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

Step 3

Multiply the new quotient term by the divisor.

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

Step 4

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

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

Step 5

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

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

Step 6

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

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

Step 7

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

$r^{2}$

$r$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

Step 8

Multiply the new quotient term by the divisor.

$r^{2}$

$r$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

Step 9

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

$r^{2}$

$r$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

Step 10

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

$r^{2}$

$r$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

$2 r^{2}$

$2 r$

Step 11

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

$r^{2}$

$r$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

$2 r^{2}$

$2 r$

$4$

Step 12

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

$r^{2}$

$r$

$2$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

$2 r^{2}$

$2 r$

$4$

Step 13

Multiply the new quotient term by the divisor.

$r^{2}$

$r$

$2$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

$2 r^{2}$

$2 r$

$4$

$2 r^{2}$

$2 r$

$4$

Step 14

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

$r^{2}$

$r$

$2$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

$2 r^{2}$

$2 r$

$4$

$2 r^{2}$

$2 r$

$4$

Step 15

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

$r^{2}$

$r$

$2$

$r^{2}$

$r$

$2$

$r^{4}$

$0 r^{3}$

$r^{2}$

$0 r$

$4$

$r^{4}$

$r^{3}$

$2 r^{2}$

$r^{3}$

$3 r^{2}$

$0 r$

$r^{3}$

$r^{2}$

$2 r$

$2 r^{2}$

$2 r$

$4$

$2 r^{2}$

$2 r$

$4$

$4 r$

$8$

Step 16

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

$r^{2} + r - 2 + \frac{- 4 r + 8}{r^{2} - r + 2}$