Precalculus Examples

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

Step 1

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

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

Step 2

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

$2 x^{2}$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 6 x^{3} + 4 x^{2}$

$2 x^{2}$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

Step 5

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

$2 x^{2}$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

Step 6

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

$2 x^{2}$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

Step 7

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

$2 x^{2}$

$6 x$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$6 x$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{3} - 18 x^{2} - 12 x$

$2 x^{2}$

$6 x$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

Step 10

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

$2 x^{2}$

$6 x$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

$13 x^{2}$

$12 x$

Step 11

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

$2 x^{2}$

$6 x$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

$13 x^{2}$

$12 x$

$6$

Step 12

Divide the highest order term in the dividend $13 x^{2}$ by the highest order term in divisor $x^{2}$ .

$2 x^{2}$

$6 x$

$13$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

$13 x^{2}$

$12 x$

$6$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{2}$

$6 x$

$13$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

$13 x^{2}$

$12 x$

$6$

$13 x^{2}$

$39 x$

$26$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{2} + 39 x + 26$

$2 x^{2}$

$6 x$

$13$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

$13 x^{2}$

$12 x$

$6$

$13 x^{2}$

$39 x$

$26$

Step 15

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

$2 x^{2}$

$6 x$

$13$

$x^{2}$

$3 x$

$2$

$2 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$6$

$2 x^{4}$

$6 x^{3}$

$4 x^{2}$

$6 x^{3}$

$5 x^{2}$

$0 x$

$6 x^{3}$

$18 x^{2}$

$12 x$

$13 x^{2}$

$12 x$

$6$

$13 x^{2}$

$39 x$

$26$

$27 x$

$20$

Step 16

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

$2 x^{2} - 6 x + 13 + \frac{- 27 x - 20}{x^{2} + 3 x + 2}$