Divide $$$u^{5}$$$ by $$$u^{3} + 1$$$

Find $$$\frac{u^{5}}{u^{3} + 1}$$$ using long division.

Write the problem in the special format (missed terms are written with zero coefficients):

$$$\begin{array}{r|r}\hline\\u^{3}+1&u^{5}+0 u^{4}+0 u^{3}+0 u^{2}+0 u+0\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{u^{5}}{u^{3}} = u^{2}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$u^{2} \left(u^{3}+1\right) = u^{5}+u^{2}$$$.

Subtract the dividend from the obtained result: $$$\left(u^{5}\right) - \left(u^{5}+u^{2}\right) = - u^{2}$$$.

Since the degree of the remainder is less than the degree of the divisor, we are done.

The resulting table is shown once more:

Therefore, $$$\frac{u^{5}}{u^{3} + 1} = u^{2} + \frac{- u^{2}}{u^{3} + 1}$$$.

$$$\frac{u^{5}}{u^{3} + 1} = u^{2} + \frac{- u^{2}}{u^{3} + 1}$$$A