Divide $$$x^{3}$$$ by $$$x^{2} - 9$$$
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Your Input
Find $$$\frac{x^{3}}{x^{2} - 9}$$$ using long division.
Solution
Write the problem in the special format (missed terms are written with zero coefficients):
$$$\begin{array}{r|r}\hline\\x^{2}-9&x^{3}+0 x^{2}+0 x+0\end{array}$$$
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: $$$x \left(x^{2}-9\right) = x^{3}- 9 x$$$.
Subtract the dividend from the obtained result: $$$\left(x^{3}\right) - \left(x^{3}- 9 x\right) = 9 x$$$.
$$\begin{array}{r|rrrr:c}&{\color{Crimson}x}&&&&\\\hline\\{\color{Magenta}x^{2}}-9&{\color{Crimson}x^{3}}&+0 x^{2}&+0 x&+0&\frac{{\color{Crimson}x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Crimson}x}\\&-\phantom{x^{3}}&&&&\\&x^{3}&+0 x^{2}&- 9 x&&{\color{Crimson}x} \left(x^{2}-9\right) = x^{3}- 9 x\\\hline\\&&&9 x&+0&\end{array}$$Since the degree of the remainder is less than the degree of the divisor, we are done.
The resulting table is shown once more:
$$\begin{array}{r|rrrr:c}&{\color{Crimson}x}&&&&\text{Hints}\\\hline\\{\color{Magenta}x^{2}}-9&{\color{Crimson}x^{3}}&+0 x^{2}&+0 x&+0&\frac{{\color{Crimson}x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Crimson}x}\\&-\phantom{x^{3}}&&&&\\&x^{3}&+0 x^{2}&- 9 x&&{\color{Crimson}x} \left(x^{2}-9\right) = x^{3}- 9 x\\\hline\\&&&9 x&+0&\end{array}$$Therefore, $$$\frac{x^{3}}{x^{2} - 9} = x + \frac{9 x}{x^{2} - 9}$$$.
Answer
$$$\frac{x^{3}}{x^{2} - 9} = x + \frac{9 x}{x^{2} - 9}$$$A