Precalculus Examples

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

Step 1

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

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 2

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

Step 4

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

Step 5

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

Step 6

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

$x^{4}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

Step 7

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

Step 8

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

Step 9

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

Step 10

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

Step 11

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

$x^{4}$

$x^{3}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 12

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 13

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 14

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 15

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 16

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 17

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 18

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 19

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 20

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 21

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 22

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 23

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 24

The expression needs to be subtracted from the dividend, so change all the signs in $x + 1$

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 25

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

$x^{4}$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

$0$

Step 26

Since the remander is $0$ , the final answer is the quotient.

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