Precalculus Examples

$\frac{2 x^{5} - 7 x^{4} - 11}{4 x^{2} - 6 x + 8}$

Step 1

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

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

Step 2

Divide the highest order term in the dividend $2 x^{5}$ by the highest order term in divisor $4 x^{2}$ .

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

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

Step 3

Multiply the new quotient term by the divisor.

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

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{5} - 3 x^{4} + 4 x^{3}$

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

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

Step 5

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

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

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

Step 6

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

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

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

Step 7

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

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

$x^{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

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

$x^{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{4} + 6 x^{3} - 8 x^{2}$

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

$x^{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

Step 10

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

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

$x^{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

Step 11

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

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

$x^{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

Step 12

Divide the highest order term in the dividend $- 10 x^{3}$ by the highest order term in divisor $4 x^{2}$ .

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

$x^{2}$

$\frac{5 x}{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

Step 13

Multiply the new quotient term by the divisor.

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

$x^{2}$

$\frac{5 x}{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{3} + 15 x^{2} - 20 x$

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

$x^{2}$

$\frac{5 x}{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

Step 15

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

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

$x^{2}$

$\frac{5 x}{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

$7 x^{2}$

$20 x$

Step 16

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

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

$x^{2}$

$\frac{5 x}{2}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

$7 x^{2}$

$20 x$

$11$

Step 17

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $4 x^{2}$ .

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

$x^{2}$

$\frac{5 x}{2}$

$\frac{7}{4}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

$7 x^{2}$

$20 x$

$11$

Step 18

Multiply the new quotient term by the divisor.

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

$x^{2}$

$\frac{5 x}{2}$

$\frac{7}{4}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

$7 x^{2}$

$20 x$

$11$

$7 x^{2}$

$\frac{21 x}{2}$

$14$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} + \frac{21 x}{2} - 14$

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

$x^{2}$

$\frac{5 x}{2}$

$\frac{7}{4}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

$7 x^{2}$

$20 x$

$11$

$7 x^{2}$

$\frac{21 x}{2}$

$14$

Step 20

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

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

$x^{2}$

$\frac{5 x}{2}$

$\frac{7}{4}$

$4 x^{2}$

$6 x$

$8$

$2 x^{5}$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$11$

$2 x^{5}$

$3 x^{4}$

$4 x^{3}$

$4 x^{4}$

$4 x^{3}$

$0 x^{2}$

$4 x^{4}$

$6 x^{3}$

$8 x^{2}$

$10 x^{3}$

$8 x^{2}$

$0 x$

$10 x^{3}$

$15 x^{2}$

$20 x$

$7 x^{2}$

$20 x$

$11$

$7 x^{2}$

$\frac{21 x}{2}$

$14$

$\frac{19 x}{2}$

$3$

Step 21

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

$\frac{x^{3}}{2} - x^{2} - \frac{5 x}{2} - \frac{7}{4} + \frac{\frac{19 x}{2} + 3}{4 x^{2} - 6 x + 8}$