Precalculus Examples

$\frac{6 x^{4} + 10 x^{3} + 13 x^{2} - 5 x + 2}{2 x^{2} - 1}$

Step 1

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

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

Step 2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $2 x^{2}$ .

$3 x^{2}$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{2}$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 0 - 3 x^{2}$

$3 x^{2}$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

Step 5

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

$3 x^{2}$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

Step 6

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

$3 x^{2}$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

Step 7

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $2 x^{2}$ .

$3 x^{2}$

$5 x$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{2}$

$5 x$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} + 0 - 5 x$

$3 x^{2}$

$5 x$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

Step 10

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

$3 x^{2}$

$5 x$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

$16 x^{2}$

$0$

Step 11

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

$3 x^{2}$

$5 x$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

$16 x^{2}$

$0$

$2$

Step 12

Divide the highest order term in the dividend $16 x^{2}$ by the highest order term in divisor $2 x^{2}$ .

$3 x^{2}$

$5 x$

$8$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

$16 x^{2}$

$0$

$2$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{2}$

$5 x$

$8$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

$16 x^{2}$

$0$

$2$

$16 x^{2}$

$0$

$8$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{2} + 0 - 8$

$3 x^{2}$

$5 x$

$8$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

$16 x^{2}$

$0$

$2$

$16 x^{2}$

$0$

$8$

Step 15

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

$3 x^{2}$

$5 x$

$8$

$2 x^{2}$

$0 x$

$1$

$6 x^{4}$

$10 x^{3}$

$13 x^{2}$

$5 x$

$2$

$6 x^{4}$

$0$

$3 x^{2}$

$10 x^{3}$

$16 x^{2}$

$5 x$

$10 x^{3}$

$0$

$5 x$

$16 x^{2}$

$0$

$2$

$16 x^{2}$

$0$

$8$

$10$

Step 16

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

$3 x^{2} + 5 x + 8 + \frac{10}{2 x^{2} - 1}$