Divide $$$x^{3}$$$ by $$$x^{2} + 1$$$

Find $$$\frac{x^{3}}{x^{2} + 1}$$$ using long division.

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

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

Step 1

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

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

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

Subtract the dividend from the obtained result: $$$\left(x^{3}\right) - \left(x^{3}+x\right) = - x$$$.

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, $$$\frac{x^{3}}{x^{2} + 1} = x + \frac{- x}{x^{2} + 1}$$$.

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