Pre-Algebra Examples

$\frac{6 x^{5} + 5 x^{4} + x^{3} - 3 x^{2} + x}{3 x + 1}$

Step 1

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

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

Step 2

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

$2 x^{4}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{4}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{5} + 2 x^{4}$

$2 x^{4}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

Step 5

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

$2 x^{4}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

Step 6

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

$2 x^{4}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

Step 7

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

$2 x^{4}$

$x^{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{4}$

$x^{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

Step 9

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

$2 x^{4}$

$x^{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

Step 10

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

$2 x^{4}$

$x^{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

Step 11

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

$2 x^{4}$

$x^{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

Step 12

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

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

Step 14

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

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

Step 15

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

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

$2 x$

Step 16

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

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

$2 x$

$0$

Step 17

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $3 x$ .

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$\frac{2}{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

$2 x$

$0$

Step 18

Multiply the new quotient term by the divisor.

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$\frac{2}{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

$2 x$

$0$

$2 x$

$\frac{2}{3}$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + \frac{2}{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$\frac{2}{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

$2 x$

$0$

$2 x$

$\frac{2}{3}$

Step 20

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

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$\frac{2}{3}$

$3 x$

$1$

$6 x^{5}$

$5 x^{4}$

$x^{3}$

$3 x^{2}$

$x$

$0$

$6 x^{5}$

$2 x^{4}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$x^{3}$

$0$

$3 x^{2}$

$x$

$3 x^{2}$

$x$

$2 x$

$0$

$2 x$

$\frac{2}{3}$

Step 21

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

$2 x^{4} + x^{3} - x + \frac{2}{3} - \frac{2}{3 (3 x + 1)}$