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\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i1)
load
(
odes
)
$
(%i2)
cubic1
:
x
^
3
-
3
*
x
^
2
-
2
*
x
+
5
=
0
;
\[\tag{cubic1}\label{cubic1}{{x}^{3}}-3{{x}^{2}}-2x+5=0\]
(%i3)
cubic2
:
x
^
3
-
3
*
x
-
1
=
0
;
\[\tag{cubic2}\label{cubic2}{{x}^{3}}-3x-1=0\]
(%i4)
solvet
(
cubic1
,
x
)
;
\[\tag{\%{}o4}\label{o4} [x=\frac{2\sqrt{5}\,\cos{\left( \frac{\operatorname{atan}\left( \frac{\sqrt{473}}{{{3}^{\frac{3}{2}}}}\right) -3\ensuremath{\pi} }{3}\right) }+\sqrt{3}}{\sqrt{3}},x=\frac{2\sqrt{5}\,\cos{\left( \frac{\operatorname{atan}\left( \frac{\sqrt{473}}{{{3}^{\frac{3}{2}}}}\right) -\ensuremath{\pi} }{3}\right) }+\sqrt{3}}{\sqrt{3}},x=\frac{2\sqrt{5}\,\cos{\left( \frac{\operatorname{atan}\left( \frac{\sqrt{473}}{{{3}^{\frac{3}{2}}}}\right) +\ensuremath{\pi} }{3}\right) }+\sqrt{3}}{\sqrt{3}}]\]
(%i5)
solvet
(
cubic2
,
x
)
;
\[\tag{\%{}o5}\label{o5} [x=2\cos{\left( \frac{\ensuremath{\pi} }{9}\right) },x=2\cos{\left( \frac{5\ensuremath{\pi} }{9}\right) },x=2\cos{\left( \frac{7\ensuremath{\pi} }{9}\right) }]\]
(%i6)
sextic
:
expand
(
cubic1
*
cubic2
)
;
\[\tag{sextic}\label{sextic}{{x}^{6}}-3{{x}^{5}}-5{{x}^{4}}+13{{x}^{3}}+9{{x}^{2}}-13x-5=0\]
(%i7)
solvet
(
sextic
,
x
)
;
\[\tag{\%{}o7}\label{o7} [x=2\cos{\left( \frac{\ensuremath{\pi} }{9}\right) },x=2\cos{\left( \frac{5\ensuremath{\pi} }{9}\right) },x=2\cos{\left( \frac{7\ensuremath{\pi} }{9}\right) },x=\frac{2\sqrt{5}\,\cos{\left( \frac{\operatorname{atan}\left( \frac{\sqrt{473}}{{{3}^{\frac{3}{2}}}}\right) -3\ensuremath{\pi} }{3}\right) }+\sqrt{3}}{\sqrt{3}},x=\frac{2\sqrt{5}\,\cos{\left( \frac{\operatorname{atan}\left( \frac{\sqrt{473}}{{{3}^{\frac{3}{2}}}}\right) -\ensuremath{\pi} }{3}\right) }+\sqrt{3}}{\sqrt{3}},x=\frac{2\sqrt{5}\,\cos{\left( \frac{\operatorname{atan}\left( \frac{\sqrt{473}}{{{3}^{\frac{3}{2}}}}\right) +\ensuremath{\pi} }{3}\right) }+\sqrt{3}}{\sqrt{3}}]\]
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