Precalculus Examples

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

Step 1

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

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

Step 2

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

$x^{3}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

Step 4

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

$x^{3}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

Step 5

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

$x^{3}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

Step 6

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

$x^{3}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

Step 7

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

$x^{3}$

$4 x^{2}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 20 x^{2}$

$x^{3}$

$4 x^{2}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

Step 10

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

$x^{3}$

$4 x^{2}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

Step 11

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

$x^{3}$

$4 x^{2}$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

Step 12

Divide the highest order term in the dividend $21 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$21 x$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

Step 13

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$21 x$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $21 x^{2} - 105 x$

$x^{3}$

$4 x^{2}$

$21 x$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

Step 15

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

$x^{3}$

$4 x^{2}$

$21 x$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

$104 x$

Step 16

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

$x^{3}$

$4 x^{2}$

$21 x$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

$104 x$

$2$

Step 17

Divide the highest order term in the dividend $104 x$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$21 x$

$104$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

$104 x$

$2$

Step 18

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$21 x$

$104$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

$104 x$

$2$

$104 x$

$520$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $104 x - 520$

$x^{3}$

$4 x^{2}$

$21 x$

$104$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

$104 x$

$2$

$104 x$

$520$

Step 20

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

$x^{3}$

$4 x^{2}$

$21 x$

$104$

$x$

$5$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$5 x^{3}$

$4 x^{3}$

$x^{2}$

$4 x^{3}$

$20 x^{2}$

$21 x^{2}$

$x$

$21 x^{2}$

$105 x$

$104 x$

$2$

$104 x$

$520$

$522$

Step 21

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

$x^{3} + 4 x^{2} + 21 x + 104 + \frac{522}{x - 5}$