Algebra Examples

\frac{x^{5} - 1}{x - 1}

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

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

$0$ .

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

Step 2

Divide the highest order term in the dividend

x^{5}

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

x

$x$ .

x^{4}

$x^{4}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

Step 3

Multiply the new quotient term by the divisor.

x^{4}

$x^{4}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

x^{5} - x^{4}

$x^{5} - x^{4}$

x^{4}

$x^{4}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

Step 5

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

x^{4}

$x^{4}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

Step 6

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

x^{4}

$x^{4}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

Step 7

Divide the highest order term in the dividend

x^{4}

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

x

$x$ .

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

Step 8

Multiply the new quotient term by the divisor.

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

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^{4}

$x^{4}$

x^{3}

$x^{3}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

Step 10

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

Step 11

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

Step 12

Divide the highest order term in the dividend

x^{3}

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

x

$x$ .

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

Step 13

Multiply the new quotient term by the divisor.

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in

x^{3} - x^{2}

$x^{3} - x^{2}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

Step 15

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

Step 16

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

Step 17

Divide the highest order term in the dividend

x^{2}

$x^{2}$ by the highest order term in divisor

x

$x$ .

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

Step 18

Multiply the new quotient term by the divisor.

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in

x^{2} - x

$x^{2} - x$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

Step 20

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

x

$x$

Step 21

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

Step 22

Divide the highest order term in the dividend

x

$x$ by the highest order term in divisor

x

$x$ .

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

Step 23

Multiply the new quotient term by the divisor.

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x

$x$

1

$1$

Step 24

The expression needs to be subtracted from the dividend, so change all the signs in

x - 1

$x - 1$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

x^{5}

$x^{5}$

x^{4}

$x^{4}$

x^{4}

$x^{4}$

0 x^{3}

$0 x^{3}$

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x^{2}

$x^{2}$

0 x

$0 x$

x^{2}

$x^{2}$

x

$x$

x

$x$

1

$1$

x

$x$

1

$1$

Step 25

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

x^{4}

$x^{4}$

x^{3}

$x^{3}$

x^{2}

$x^{2}$

x

$x$

1

$1$

x

$x$

1

$1$

x^{5}

$x^{5}$

0 x^{4}

$0 x^{4}$

0 x^{3}

$0 x^{3}$

0 x^{2}

$0 x^{2}$

0 x

$0 x$

1

$1$

$x^{5}$

$x^{4}$

$0 x^{3}$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

$0$

Step 26

Since the remander is

$0$ , the final answer is the quotient.

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