Find the Remainder (x^5+5x^3-9x+1)/(x^2+6)

$\frac{x^{5} + 5 x^{3} - 9 x + 1}{x^{2} + 6}$

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 $0$ .

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

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

$x^{3}$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} + 0 + 6 x^{3}$

$x^{3}$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

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

$x^{3}$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

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

$x^{3}$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

$0 x^{2}$

$9 x$

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

$x^{3}$

$0 x^{2}$

$x$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

$0 x^{2}$

$9 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$x$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

$0 x^{2}$

$9 x$

$x^{3}$

$0$

$6 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + 0 - 6 x$

$x^{3}$

$0 x^{2}$

$x$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

$0 x^{2}$

$9 x$

$x^{3}$

$0$

$6 x$

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

$x^{3}$

$0 x^{2}$

$x$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

$0 x^{2}$

$9 x$

$x^{3}$

$0$

$6 x$

$3 x$

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

$x^{3}$

$0 x^{2}$

$x$

$x^{2}$

$0 x$

$6$

$x^{5}$

$0 x^{4}$

$5 x^{3}$

$0 x^{2}$

$9 x$

$1$

$x^{5}$

$0$

$6 x^{3}$

$x^{3}$

$0 x^{2}$

$9 x$

$x^{3}$

$0$

$6 x$

$3 x$

$1$

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

$x^{3} - x + \frac{- 3 x + 1}{x^{2} + 6}$

Step 2

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

$- 3 x + 1$

Find the Remainder (x^5+5x^3-9x+1)/(x^2+6)

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