Pre-Algebra Examples

$\frac{12 x^{5} - 35 x^{4} + 31 x^{3} - 11 x^{2} + 4 x - 6}{3 x^{2} - 5 x + 3}$

Step 1

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

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

Step 2

Divide the highest order term in the dividend $12 x^{5}$ by the highest order term in divisor $3 x^{2}$ .

$4 x^{3}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

Step 3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{5} - 20 x^{4} + 12 x^{3}$

$4 x^{3}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

Step 5

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

$4 x^{3}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

Step 6

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

$4 x^{3}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

Step 7

Divide the highest order term in the dividend $- 15 x^{4}$ by the highest order term in divisor $3 x^{2}$ .

$4 x^{3}$

$5 x^{2}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$5 x^{2}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{4} + 25 x^{3} - 15 x^{2}$

$4 x^{3}$

$5 x^{2}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

Step 10

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

$4 x^{3}$

$5 x^{2}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

Step 11

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

$4 x^{3}$

$5 x^{2}$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

Step 12

Divide the highest order term in the dividend $- 6 x^{3}$ by the highest order term in divisor $3 x^{2}$ .

$4 x^{3}$

$5 x^{2}$

$2 x$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

Step 13

Multiply the new quotient term by the divisor.

$4 x^{3}$

$5 x^{2}$

$2 x$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{3} + 10 x^{2} - 6 x$

$4 x^{3}$

$5 x^{2}$

$2 x$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

Step 15

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

$4 x^{3}$

$5 x^{2}$

$2 x$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

$6 x^{2}$

$10 x$

Step 16

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

$4 x^{3}$

$5 x^{2}$

$2 x$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

$6 x^{2}$

$10 x$

$6$

Step 17

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

$4 x^{3}$

$5 x^{2}$

$2 x$

$2$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

$6 x^{2}$

$10 x$

$6$

Step 18

Multiply the new quotient term by the divisor.

$4 x^{3}$

$5 x^{2}$

$2 x$

$2$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

$6 x^{2}$

$10 x$

$6$

$6 x^{2}$

$10 x$

$6$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 10 x - 6$

$4 x^{3}$

$5 x^{2}$

$2 x$

$2$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

$6 x^{2}$

$10 x$

$6$

$6 x^{2}$

$10 x$

$6$

Step 20

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

$4 x^{3}$

$5 x^{2}$

$2 x$

$2$

$3 x^{2}$

$5 x$

$3$

$12 x^{5}$

$35 x^{4}$

$31 x^{3}$

$11 x^{2}$

$4 x$

$6$

$12 x^{5}$

$20 x^{4}$

$12 x^{3}$

$15 x^{4}$

$19 x^{3}$

$11 x^{2}$

$15 x^{4}$

$25 x^{3}$

$15 x^{2}$

$6 x^{3}$

$4 x^{2}$

$4 x$

$6 x^{3}$

$10 x^{2}$

$6 x$

$6 x^{2}$

$10 x$

$6$

$6 x^{2}$

$10 x$

$6$

$0$

Step 21

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

$4 x^{3} - 5 x^{2} - 2 x - 2$