Algebra Examples

$(6 y^{4} + 15 y^{3} - 28 y^{2} - 6) \div (y + 2)$

Step 1

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

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

Step 2

Divide the highest order term in the dividend $6 y^{4}$ by the highest order term in divisor $y$ .

$6 y^{3}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

Step 3

Multiply the new quotient term by the divisor.

$6 y^{3}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 y^{4} + 12 y^{3}$

$6 y^{3}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

Step 5

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

$6 y^{3}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

Step 6

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

$6 y^{3}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

Step 7

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

$6 y^{3}$

$3 y^{2}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

Step 8

Multiply the new quotient term by the divisor.

$6 y^{3}$

$3 y^{2}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

Step 9

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

$6 y^{3}$

$3 y^{2}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

Step 10

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

$6 y^{3}$

$3 y^{2}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

Step 11

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

$6 y^{3}$

$3 y^{2}$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

Step 12

Divide the highest order term in the dividend $- 34 y^{2}$ by the highest order term in divisor $y$ .

$6 y^{3}$

$3 y^{2}$

$34 y$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

Step 13

Multiply the new quotient term by the divisor.

$6 y^{3}$

$3 y^{2}$

$34 y$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 34 y^{2} - 68 y$

$6 y^{3}$

$3 y^{2}$

$34 y$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

Step 15

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

$6 y^{3}$

$3 y^{2}$

$34 y$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

Step 16

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

$6 y^{3}$

$3 y^{2}$

$34 y$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

$6$

Step 17

Divide the highest order term in the dividend $68 y$ by the highest order term in divisor $y$ .

$6 y^{3}$

$3 y^{2}$

$34 y$

$68$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

$6$

Step 18

Multiply the new quotient term by the divisor.

$6 y^{3}$

$3 y^{2}$

$34 y$

$68$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

$6$

$68 y$

$136$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $68 y + 136$

$6 y^{3}$

$3 y^{2}$

$34 y$

$68$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

$6$

$68 y$

$136$

Step 20

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

$6 y^{3}$

$3 y^{2}$

$34 y$

$68$

$y$

$2$

$6 y^{4}$

$15 y^{3}$

$28 y^{2}$

$0 y$

$6$

$6 y^{4}$

$12 y^{3}$

$3 y^{3}$

$28 y^{2}$

$3 y^{3}$

$6 y^{2}$

$34 y^{2}$

$0 y$

$34 y^{2}$

$68 y$

$6$

$68 y$

$136$

$142$

Step 21

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

$6 y^{3} + 3 y^{2} - 34 y + 68 - \frac{142}{y + 2}$