Find the Remainder (3x^5-x^4+4x^2-51)/(x^2+1)

$\frac{3 x^{5} - x^{4} + 4 x^{2} - 51}{x^{2} + 1}$

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 $0$ .

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

Divide the highest order term in the dividend $3 x^{5}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{5} + 0 + 3 x^{3}$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

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

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

Divide the highest order term in the dividend $- x^{4}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{4} + 0 - x^{2}$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

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

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

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

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

Divide the highest order term in the dividend $- 3 x^{3}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

$x^{2}$

$3 x$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$3 x$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} + 0 - 3 x$

$3 x^{3}$

$x^{2}$

$3 x$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

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

$3 x^{3}$

$x^{2}$

$3 x$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

$5 x^{2}$

$3 x$

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

$3 x^{3}$

$x^{2}$

$3 x$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

$5 x^{2}$

$3 x$

$51$

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

$x^{2}$

$3 x$

$5$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

$5 x^{2}$

$3 x$

$51$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$3 x$

$5$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

$5 x^{2}$

$3 x$

$51$

$5 x^{2}$

$0$

$5$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 0 + 5$

$3 x^{3}$

$x^{2}$

$3 x$

$5$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

$5 x^{2}$

$3 x$

$51$

$5 x^{2}$

$0$

$5$

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

$3 x^{3}$

$x^{2}$

$3 x$

$5$

$x^{2}$

$0 x$

$1$

$3 x^{5}$

$x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$51$

$3 x^{5}$

$0$

$3 x^{3}$

$x^{4}$

$3 x^{3}$

$4 x^{2}$

$x^{4}$

$0$

$x^{2}$

$3 x^{3}$

$5 x^{2}$

$0 x$

$3 x^{3}$

$0$

$3 x$

$5 x^{2}$

$3 x$

$51$

$5 x^{2}$

$0$

$5$

$3 x$

$56$

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

$3 x^{3} - x^{2} - 3 x + 5 + \frac{3 x - 56}{x^{2} + 1}$

Step 2

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

$3 x - 56$

Find the Remainder (3x^5-x^4+4x^2-51)/(x^2+1)

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