Find the Remainder (-13x^9+6x^5+12x)/(x^6)

$\frac{- 13 x^{9} + 6 x^{5} + 12 x}{x^{6}}$

To calculate the remainder, first divide the polynomials.

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

$x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

–

$13 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$6 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$0$

Divide the highest order term in the dividend $- 13 x^{9}$ by the highest order term in divisor $x^{6}$ .

–

$13 x^{3}$

$x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

–

$13 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$6 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$0$

Multiply the new quotient term by the divisor.

–

$13 x^{3}$

$x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

–

$13 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$6 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$0$

–

$13 x^{9}$

$0$

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{9} + 0 + 0 + 0 + 0 + 0 + 0$

–

$13 x^{3}$

$x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

–

$13 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$6 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$0$

$13 x^{9}$

–

$0$

–

$0$

–

$0$

–

$0$

–

$0$

–

$0$

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

–

$13 x^{3}$

$x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

–

$13 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$6 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$0$

$13 x^{9}$

–

$0$

–

$0$

–

$0$

–

$0$

–

$0$

–

$0$

$6 x^{5}$

$0$

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

–

$13 x^{3}$

$x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

–

$13 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$6 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$0$

$13 x^{9}$

–

$0$

–

$0$

–

$0$

–

$0$

–

$0$

–

$0$

$6 x^{5}$

$0$

$0 x^{2}$

$12 x$

$0$

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

$- 13 x^{3} + \frac{6 x^{5} + 12 x}{x^{6}}$

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

$6 x^{5} + 12 x$

Find the Remainder (-13x^9+6x^5+12x)/(x^6)

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