Pre-Algebra Examples

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

Step 1

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

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

Step 2

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

$x^{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

Step 4

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

$x^{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

Step 5

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

$x^{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

Step 6

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

$x^{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

Step 7

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

$x^{2}$

$2 x$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

Step 8

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

Step 9

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

$x^{2}$

$2 x$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

Step 10

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

$x^{2}$

$2 x$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

$x^{2}$

$3 x$

Step 11

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

$x^{2}$

$2 x$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

$x^{2}$

$3 x$

$1$

Step 12

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

$x^{2}$

$2 x$

$\frac{1}{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

$x^{2}$

$3 x$

$1$

Step 13

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$\frac{1}{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

$x^{2}$

$3 x$

$1$

$x^{2}$

$\frac{x}{2}$

$\frac{1}{2}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + \frac{x}{2} + \frac{1}{2}$

$x^{2}$

$2 x$

$\frac{1}{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

$x^{2}$

$3 x$

$1$

$x^{2}$

$\frac{x}{2}$

$\frac{1}{2}$

Step 15

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

$x^{2}$

$2 x$

$\frac{1}{2}$

$2 x^{2}$

$x$

$1$

$2 x^{4}$

$3 x^{3}$

$0 x^{2}$

$x$

$1$

$2 x^{4}$

$x^{3}$

$x^{2}$

$4 x^{3}$

$x^{2}$

$x$

$4 x^{3}$

$2 x^{2}$

$2 x$

$x^{2}$

$3 x$

$1$

$x^{2}$

$\frac{x}{2}$

$\frac{1}{2}$

$\frac{5 x}{2}$

$\frac{1}{2}$

Step 16

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

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