Divide Using Long Polynomial Division (5t^4-10t^2+6)/(t+2)

$\frac{5 t^{4} - 10 t^{2} + 6}{t + 2}$

Step 1

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

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

Step 2

Divide the highest order term in the dividend $5 t^{4}$ by the highest order term in divisor $t$ .

$5 t^{3}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

Step 3

Multiply the new quotient term by the divisor.

$5 t^{3}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 t^{4} + 10 t^{3}$

$5 t^{3}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

Step 5

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

$5 t^{3}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

Step 6

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

$5 t^{3}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

Step 7

Divide the highest order term in the dividend $- 10 t^{3}$ by the highest order term in divisor $t$ .

$5 t^{3}$

$10 t^{2}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

Step 8

Multiply the new quotient term by the divisor.

$5 t^{3}$

$10 t^{2}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 t^{3} - 20 t^{2}$

$5 t^{3}$

$10 t^{2}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

Step 10

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

$5 t^{3}$

$10 t^{2}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

Step 11

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

$5 t^{3}$

$10 t^{2}$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

Step 12

Divide the highest order term in the dividend $10 t^{2}$ by the highest order term in divisor $t$ .

$5 t^{3}$

$10 t^{2}$

$10 t$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

Step 13

Multiply the new quotient term by the divisor.

$5 t^{3}$

$10 t^{2}$

$10 t$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $10 t^{2} + 20 t$

$5 t^{3}$

$10 t^{2}$

$10 t$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

Step 15

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

$5 t^{3}$

$10 t^{2}$

$10 t$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

Step 16

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

$5 t^{3}$

$10 t^{2}$

$10 t$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

$6$

Step 17

Divide the highest order term in the dividend $- 20 t$ by the highest order term in divisor $t$ .

$5 t^{3}$

$10 t^{2}$

$10 t$

$20$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

$6$

Step 18

Multiply the new quotient term by the divisor.

$5 t^{3}$

$10 t^{2}$

$10 t$

$20$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

$6$

$20 t$

$40$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 t - 40$

$5 t^{3}$

$10 t^{2}$

$10 t$

$20$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

$6$

$20 t$

$40$

Step 20

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

$5 t^{3}$

$10 t^{2}$

$10 t$

$20$

$t$

$2$

$5 t^{4}$

$0 t^{3}$

$10 t^{2}$

$0 t$

$6$

$5 t^{4}$

$10 t^{3}$

$10 t^{2}$

$10 t^{3}$

$20 t^{2}$

$10 t^{2}$

$0 t$

$10 t^{2}$

$20 t$

$6$

$20 t$

$40$

$46$

Step 21

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

$5 t^{3} - 10 t^{2} + 10 t - 20 + \frac{46}{t + 2}$

Divide Using Long Polynomial Division (5t^4-10t^2+6)/(t+2)

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