Algebra Examples

$(10 x^{6} + 20 x^{4} - 15 x^{2}) \div 5 x^{2}$

Step 1

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

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $10 x^{6}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{4}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{4}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{6} + 0 + 0$

$2 x^{4}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

Step 5

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

$2 x^{4}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

Step 6

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

$2 x^{4}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

Step 7

Divide the highest order term in the dividend $20 x^{4}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{4} + 0 + 0$

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

Step 10

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

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

$15 x^{2}$

Step 11

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

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

$15 x^{2}$

$0 x$

$0$

Step 12

Divide the highest order term in the dividend $- 15 x^{2}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$3$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

$15 x^{2}$

$0 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$3$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

$15 x^{2}$

$0 x$

$0$

$15 x^{2}$

$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} + 0 + 0$

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$3$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

$15 x^{2}$

$0 x$

$0$

$15 x^{2}$

$0$

Step 15

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

$2 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$3$

$5 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0 x^{5}$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$0 x$

$0$

$10 x^{6}$

$0$

$20 x^{4}$

$0 x^{3}$

$15 x^{2}$

$20 x^{4}$

$0$

$15 x^{2}$

$0 x$

$0$

$15 x^{2}$

$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

$2 x^{4} + 0 x^{3} + 4 x^{2} + 0 x - 3$