View Single Post
Old 2020-12-13, 21:06   #35
alpertron
 
alpertron's Avatar
 
Aug 2002
Buenos Aires, Argentina

22·3·113 Posts
Default

I've just added TeX output to my polynomial factorization calculator located at https://www.alpertron.com.ar/POLFACT.HTM

For example, the roots of x17 + 1 are:

[math]\begin{array}{l}
\bullet\,\,x_{1} = -1\\
\bullet\,\,x_{2} = \cos{\frac{\pi }{17}} + i \sin{\frac{\pi }{17}} = \frac{1}{16}\left(1-\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34-2\sqrt{17}-2\sqrt{34-2\sqrt{17}}-4\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{3} = \cos{ \frac{3\pi }{17}} + i \sin{\frac{3\pi }{17}} = \frac{1}{16}\left(1+\sqrt{17}+\sqrt{34+2\sqrt{17}}+2\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34+2\sqrt{17}-2\sqrt{34+2\sqrt{17}}-4\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{4} = \cos{ \frac{5\pi }{17}} + i \sin{\frac{5\pi }{17}} = \frac{1}{16}\left(1+\sqrt{17}+\sqrt{34+2\sqrt{17}}-2\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34+2\sqrt{17}-2\sqrt{34+2\sqrt{17}}+4\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{5} = \cos{ \frac{7\pi }{17}} + i \sin{\frac{7\pi }{17}} = \frac{1}{16}\left(1+\sqrt{17}-\sqrt{34+2\sqrt{17}}+2\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34+2\sqrt{17}+2\sqrt{34+2\sqrt{17}}+4\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{6} = \cos{ \frac{9\pi }{17}} + i \sin{\frac{9\pi }{17}} = -\frac{1}{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}-2\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34-2\sqrt{17}+2\sqrt{34-2\sqrt{17}}+4\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{7} = \cos{ \frac{11\pi }{17}} + i \sin{\frac{11\pi }{17}} = -\frac{1}{16}\left(-1-\sqrt{17}+\sqrt{34+2\sqrt{17}}+2\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34+2\sqrt{17}+2\sqrt{34+2\sqrt{17}}-4\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{8} = \cos{ \frac{13\pi }{17}} + i \sin{\frac{13\pi }{17}} = -\frac{1}{16}\left(-1+\sqrt{17}-\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34-2\sqrt{17}-2\sqrt{34-2\sqrt{17}}+4\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{9} = \cos{ \frac{15\pi }{17}} + i \sin{\frac{15\pi }{17}} = -\frac{1}{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}\right) + \frac{i}{8}\sqrt{34-2\sqrt{17}+2\sqrt{34-2\sqrt{17}}-4\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{10} = \cos{ \frac{19\pi }{17}} + i \sin{\frac{19\pi }{17}} = -\frac{1}{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34-2\sqrt{17}+2\sqrt{34-2\sqrt{17}}-4\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{11} = \cos{ \frac{21\pi }{17}} + i \sin{\frac{21\pi }{17}} = -\frac{1}{16}\left(-1+\sqrt{17}-\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34-2\sqrt{17}-2\sqrt{34-2\sqrt{17}}+4\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{12} = \cos{ \frac{23\pi }{17}} + i \sin{\frac{23\pi }{17}} = -\frac{1}{16}\left(-1-\sqrt{17}+\sqrt{34+2\sqrt{17}}+2\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34+2\sqrt{17}+2\sqrt{34+2\sqrt{17}}-4\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{13} = \cos{ \frac{25\pi }{17}} + i \sin{\frac{25\pi }{17}} = -\frac{1}{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}-2\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34-2\sqrt{17}+2\sqrt{34-2\sqrt{17}}+4\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}}\\
\bullet\,\,x_{14} = \cos{ \frac{27\pi }{17}} + i \sin{\frac{27\pi }{17}} = \frac{1}{16}\left(1+\sqrt{17}-\sqrt{34+2\sqrt{17}}+2\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34+2\sqrt{17}+2\sqrt{34+2\sqrt{17}}+4\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{15} = \cos{ \frac{29\pi }{17}} + i \sin{\frac{29\pi }{17}} = \frac{1}{16}\left(1+\sqrt{17}+\sqrt{34+2\sqrt{17}}-2\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34+2\sqrt{17}-2\sqrt{34+2\sqrt{17}}+4\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{16} = \cos{ \frac{31\pi }{17}} + i \sin{\frac{31\pi }{17}} = \frac{1}{16}\left(1+\sqrt{17}+\sqrt{34+2\sqrt{17}}+2\sqrt{17-3\sqrt{17}-\sqrt{170-38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34+2\sqrt{17}-2\sqrt{34+2\sqrt{17}}-4\sqrt{17-3\sqrt{17}+\sqrt{170-38\sqrt{17}}}}\\
\bullet\,\,x_{17} = \cos{ \frac{33\pi }{17}} + i \sin{\frac{33\pi }{17}} = \frac{1}{16}\left(1-\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}+\sqrt{170+38\sqrt{17}}}\right)-\frac{i}{8}\sqrt{34-2\sqrt{17}-2\sqrt{34-2\sqrt{17}}-4\sqrt{17+3\sqrt{17}-\sqrt{170+38\sqrt{17}}}}\\
\end{array}[/math]

I had to change a lot of code to do this, so it is possible that there are some errors. So I will appreciate if you post here the error(s) you can find.
alpertron is offline   Reply With Quote