Pre-Algebra Examples

$\frac{3 x^{7} - 5 x^{6} + 9 x^{4} + 2 x^{3} - 23}{2 x^{3} + 50}$

Step 1

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

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

Step 2

Divide the highest order term in the dividend $3 x^{7}$ by the highest order term in divisor $2 x^{3}$ .

$\frac{3 x^{4}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

Step 3

Multiply the new quotient term by the divisor.

$\frac{3 x^{4}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{7} + 0 + 0 + 75 x^{4}$

$\frac{3 x^{4}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

Step 5

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

$\frac{3 x^{4}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

Step 6

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

$\frac{3 x^{4}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

Step 7

Divide the highest order term in the dividend $- 5 x^{6}$ by the highest order term in divisor $2 x^{3}$ .

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{6} + 0 + 0 - 125 x^{3}$

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

Step 10

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

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

Step 11

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

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

Step 12

Divide the highest order term in the dividend $- 66 x^{4}$ by the highest order term in divisor $2 x^{3}$ .

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

Step 13

Multiply the new quotient term by the divisor.

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 66 x^{4} + 0 + 0 - 1650 x$

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

Step 15

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

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

$127 x^{3}$

$0$

$1650 x$

Step 16

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

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

$127 x^{3}$

$0$

$1650 x$

$23$

Step 17

Divide the highest order term in the dividend $127 x^{3}$ by the highest order term in divisor $2 x^{3}$ .

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$\frac{127}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

$127 x^{3}$

$0$

$1650 x$

$23$

Step 18

Multiply the new quotient term by the divisor.

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$\frac{127}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

$127 x^{3}$

$0$

$1650 x$

$23$

$127 x^{3}$

$0$

$3175$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $127 x^{3} + 0 + 0 + 3175$

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$\frac{127}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

$127 x^{3}$

$0$

$1650 x$

$23$

$127 x^{3}$

$0$

$3175$

Step 20

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

$\frac{3 x^{4}}{2}$

$\frac{5 x^{3}}{2}$

$0 x^{2}$

$33 x$

$\frac{127}{2}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$50$

$3 x^{7}$

$5 x^{6}$

$0 x^{5}$

$9 x^{4}$

$2 x^{3}$

$0 x^{2}$

$0 x$

$23$

$3 x^{7}$

$0$

$75 x^{4}$

$5 x^{6}$

$0$

$66 x^{4}$

$2 x^{3}$

$0 x^{2}$

$5 x^{6}$

$0$

$125 x^{3}$

$66 x^{4}$

$127 x^{3}$

$0 x^{2}$

$0 x$

$66 x^{4}$

$0$

$1650 x$

$127 x^{3}$

$0$

$1650 x$

$23$

$127 x^{3}$

$0$

$3175$

$1650 x$

$3198$

Step 21

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

$\frac{3 x^{4}}{2} - \frac{5 x^{3}}{2} - 33 x + \frac{127}{2} + \frac{1650 x - 3198}{2 x^{3} + 50}$