Find the Remainder (x^4+2x^3-3x^2+x+1)/(x-3)

$\frac{(x^{4} + 2 x^{3} - 3 x^{2} + x + 1)}{(x - 3)}$

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 $0$ .

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 15 x^{2}$

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

Divide the highest order term in the dividend $12 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} - 36 x$

$x^{3}$

$5 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

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

$x^{3}$

$5 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$37 x$

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

$x^{3}$

$5 x^{2}$

$12 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$37 x$

$1$

Divide the highest order term in the dividend $37 x$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$12 x$

$37$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$37 x$

$1$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$12 x$

$37$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$37 x$

$1$

$37 x$

$111$

The expression needs to be subtracted from the dividend, so change all the signs in $37 x - 111$

$x^{3}$

$5 x^{2}$

$12 x$

$37$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$37 x$

$1$

$37 x$

$111$

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

$x^{3}$

$5 x^{2}$

$12 x$

$37$

$x$

$3$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$3 x^{2}$

$5 x^{3}$

$15 x^{2}$

$12 x^{2}$

$x$

$12 x^{2}$

$36 x$

$37 x$

$1$

$37 x$

$111$

$112$

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

$x^{3} + 5 x^{2} + 12 x + 37 + \frac{112}{x - 3}$

Step 2

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

$112$

Find the Remainder (x^4+2x^3-3x^2+x+1)/(x-3)

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