Algebra Examples

$(- 15 x^{5} + 10 x^{4} + 15 x^{2} - 37 x + 18) \div (3 x - 2)$

Step 1

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

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

Step 2

Divide the highest order term in the dividend $- 15 x^{5}$ by the highest order term in divisor $3 x$ .

$5 x^{4}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{4}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{5} + 10 x^{4}$

$5 x^{4}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

Step 5

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

$5 x^{4}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

Step 6

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

$5 x^{4}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

Step 7

Divide the highest order term in the dividend $0 x^{3}$ by the highest order term in divisor $3 x$ .

$5 x^{4}$

$0 x^{3}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{4}$

$0 x^{3}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $0 + 0$

$5 x^{4}$

$0 x^{3}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

Step 10

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

$5 x^{4}$

$0 x^{3}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

Step 11

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

$5 x^{4}$

$0 x^{3}$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

Step 12

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

$5 x^{4}$

$0 x^{3}$

$5 x$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{4}$

$0 x^{3}$

$5 x$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{2} - 10 x$

$5 x^{4}$

$0 x^{3}$

$5 x$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

Step 15

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

$5 x^{4}$

$0 x^{3}$

$5 x$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

$27 x$

Step 16

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

$5 x^{4}$

$0 x^{3}$

$5 x$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

$27 x$

$18$

Step 17

Divide the highest order term in the dividend $- 27 x$ by the highest order term in divisor $3 x$ .

$5 x^{4}$

$0 x^{3}$

$5 x$

$9$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

$27 x$

$18$

Step 18

Multiply the new quotient term by the divisor.

$5 x^{4}$

$0 x^{3}$

$5 x$

$9$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

$27 x$

$18$

$27 x$

$18$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 27 x + 18$

$5 x^{4}$

$0 x^{3}$

$5 x$

$9$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

$27 x$

$18$

$27 x$

$18$

Step 20

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

$5 x^{4}$

$0 x^{3}$

$5 x$

$9$

$3 x$

$2$

$15 x^{5}$

$10 x^{4}$

$0 x^{3}$

$15 x^{2}$

$37 x$

$18$

$15 x^{5}$

$10 x^{4}$

$0$

$0 x^{3}$

$15 x^{2}$

$0$

$15 x^{2}$

$37 x$

$15 x^{2}$

$10 x$

$27 x$

$18$

$27 x$

$18$

$0$

Step 21

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

$- 5 x^{4} + 0 x^{3} + 5 x - 9$