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Divide $$$u^{2}$$$ by $$$1 - u^{2}$$$
Related calculators: Synthetic Division Calculator, Long Division Calculator
Your Input
Find $$$\frac{u^{2}}{1 - u^{2}}$$$ using long division.
Solution
Write the problem in the special format (missed terms are written with zero coefficients):
$$$\begin{array}{r|r}\hline\\- u^{2}+1&u^{2}+0 u+0\end{array}$$$
Step 1
Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{u^{2}}{- u^{2}} = -1$$$.
Write down the calculated result in the upper part of the table.
Multiply it by the divisor: $$$- \left(- u^{2}+1\right) = u^{2}-1$$$.
Subtract the dividend from the obtained result: $$$\left(u^{2}\right) - \left(u^{2}-1\right) = 1$$$.
$$\begin{array}{r|rrr:c}&{\color{Brown}-1}&&&\\\hline\\{\color{Magenta}- u^{2}}+1&{\color{Brown}u^{2}}&+0 u&+0&\frac{{\color{Brown}u^{2}}}{{\color{Magenta}- u^{2}}} = {\color{Brown}-1}\\&-\phantom{u^{2}}&&&\\&u^{2}&+0 u&-1&{\color{Brown}-1} \left(- u^{2}+1\right) = u^{2}-1\\\hline\\&&&1&\end{array}$$Since the degree of the remainder is less than the degree of the divisor, we are done.
The resulting table is shown once more:
$$\begin{array}{r|rrr:c}&{\color{Brown}-1}&&&\text{Hints}\\\hline\\{\color{Magenta}- u^{2}}+1&{\color{Brown}u^{2}}&+0 u&+0&\frac{{\color{Brown}u^{2}}}{{\color{Magenta}- u^{2}}} = {\color{Brown}-1}\\&-\phantom{u^{2}}&&&\\&u^{2}&+0 u&-1&{\color{Brown}-1} \left(- u^{2}+1\right) = u^{2}-1\\\hline\\&&&1&\end{array}$$Therefore, $$$\frac{u^{2}}{1 - u^{2}} = -1 + \frac{1}{1 - u^{2}}$$$.
Answer
$$$\frac{u^{2}}{1 - u^{2}} = -1 + \frac{1}{1 - u^{2}}$$$A