Algebra Examples

$(15 x^{5} - 2 x^{4} + 12 x^{3} - 4 x^{2} + x - 3) \div (3 x^{3} + 2 x^{2} + 7 x - 2)$

Step 1

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

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

Step 2

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

$5 x^{2}$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{2}$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{5} + 10 x^{4} + 35 x^{3} - 10 x^{2}$

$5 x^{2}$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

Step 5

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

$5 x^{2}$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

Step 6

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

$5 x^{2}$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

Step 7

Divide the highest order term in the dividend $- 12 x^{4}$ by the highest order term in divisor $3 x^{3}$ .

$5 x^{2}$

$4 x$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{2}$

$4 x$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{4} - 8 x^{3} - 28 x^{2} + 8 x$

$5 x^{2}$

$4 x$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

Step 10

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

$5 x^{2}$

$4 x$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

$15 x^{3}$

$34 x^{2}$

$7 x$

Step 11

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

$5 x^{2}$

$4 x$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

$15 x^{3}$

$34 x^{2}$

$7 x$

$3$

Step 12

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

$5 x^{2}$

$4 x$

$5$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

$15 x^{3}$

$34 x^{2}$

$7 x$

$3$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{2}$

$4 x$

$5$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

$15 x^{3}$

$34 x^{2}$

$7 x$

$3$

$15 x^{3}$

$10 x^{2}$

$35 x$

$10$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{3} - 10 x^{2} - 35 x + 10$

$5 x^{2}$

$4 x$

$5$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

$15 x^{3}$

$34 x^{2}$

$7 x$

$3$

$15 x^{3}$

$10 x^{2}$

$35 x$

$10$

Step 15

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

$5 x^{2}$

$4 x$

$5$

$3 x^{3}$

$2 x^{2}$

$7 x$

$2$

$15 x^{5}$

$2 x^{4}$

$12 x^{3}$

$4 x^{2}$

$x$

$3$

$15 x^{5}$

$10 x^{4}$

$35 x^{3}$

$10 x^{2}$

$12 x^{4}$

$23 x^{3}$

$6 x^{2}$

$x$

$12 x^{4}$

$8 x^{3}$

$28 x^{2}$

$8 x$

$15 x^{3}$

$34 x^{2}$

$7 x$

$3$

$15 x^{3}$

$10 x^{2}$

$35 x$

$10$

$44 x^{2}$

$28 x$

$13$

Step 16

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

$5 x^{2} - 4 x - 5 + \frac{44 x^{2} + 28 x - 13}{3 x^{3} + 2 x^{2} + 7 x - 2}$