Find the Remainder x^4-2 , x-1

$x^{4} - 2$ , $x - 1$

Step 1

Divide the higher order polynomial by the other polynomial in order to find the remainder.

$\frac{x^{4} - 2}{x - 1}$

Step 2

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

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

Step 3

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

Step 4

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

Step 5

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

Step 6

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

Step 7

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 8

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 9

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 10

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 11

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 12

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 13

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 14

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 15

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 16

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 17

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$2$

Step 18

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$2$

Step 19

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$2$

$x$

$1$

Step 20

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$2$

$x$

$1$

Step 21

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$2$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$2$

$x$

$1$

Step 22

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

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

Step 23

The remainder is the part of the answer that is left after the division by $x - 1$ is complete.

$- 1$

Find the Remainder x^4-2 , x-1

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