Find the Remainder (x^5+x^3-x)/(x-3)

$\frac{(x^{5} + x^{3} - x)}{(x - 3)}$

Step 1

To calculate the remainder, first divide the polynomials.

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

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

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

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

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

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

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

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

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

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

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

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} - 9 x^{3}$

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

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

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

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

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

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

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} - 30 x^{2}$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

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

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

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

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

Divide the highest order term in the dividend $30 x^{2}$ by the highest order term in divisor $x$ .

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

The expression needs to be subtracted from the dividend, so change all the signs in $30 x^{2} - 90 x$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

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

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

$89 x$

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

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

$89 x$

$0$

Divide the highest order term in the dividend $89 x$ by the highest order term in divisor $x$ .

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$89$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

$89 x$

$0$

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$89$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

$89 x$

$0$

$89 x$

$267$

The expression needs to be subtracted from the dividend, so change all the signs in $89 x - 267$

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$89$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

$89 x$

$0$

$89 x$

$267$

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

$x^{4}$

$3 x^{3}$

$10 x^{2}$

$30 x$

$89$

$x$

$3$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{5}$

$3 x^{4}$

$x^{3}$

$3 x^{4}$

$9 x^{3}$

$10 x^{3}$

$0 x^{2}$

$10 x^{3}$

$30 x^{2}$

$x$

$30 x^{2}$

$90 x$

$89 x$

$0$

$89 x$

$267$

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

$x^{4} + 3 x^{3} + 10 x^{2} + 30 x + 89 + \frac{267}{x - 3}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$267$

Find the Remainder (x^5+x^3-x)/(x-3)

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