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