Уравнением пятой степени называют уравнение вида:
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
=
0
{\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0}
править
Корни уравнения пятой степени
x
1
,
x
2
,
x
3
,
x
4
,
x
5
{\displaystyle x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5}}
a
,
b
,
c
,
d
,
e
,
f
{\displaystyle a,\,b,\,c,\,d,\,e,\,f}
x
1
+
x
2
+
x
3
+
x
4
+
x
5
=
−
b
a
,
{\displaystyle x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=-{\frac {b}{a}},}
x
1
x
2
+
x
1
x
3
+
x
1
x
4
+
x
1
x
5
+
x
2
x
3
+
x
2
x
4
+
x
2
x
5
+
x
3
x
4
+
x
3
x
5
+
x
4
x
5
=
c
a
,
{\displaystyle x_{1}\,x_{2}+x_{1}\,x_{3}+x_{1}\,x_{4}+x_{1}\,x_{5}+x_{2}\,x_{3}+x_{2}\,x_{4}+x_{2}\,x_{5}+x_{3}\,x_{4}+x_{3}\,x_{5}+x_{4}\,x_{5}={\frac {c}{a}},}
x
1
x
2
x
3
+
x
1
x
2
x
4
+
x
1
x
2
x
5
+
x
1
x
3
x
4
+
x
1
x
3
x
5
+
x
1
x
4
x
5
+
x
2
x
3
x
4
+
x
2
x
3
x
5
+
x
2
x
4
x
5
+
x
3
x
4
x
5
=
−
d
a
,
{\displaystyle x_{1}\,x_{2}\,x_{3}+x_{1}\,x_{2}\,x_{4}+x_{1}\,x_{2}\,x_{5}+x_{1}\,x_{3}\,x_{4}+x_{1}\,x_{3}\,x_{5}+x_{1}\,x_{4}\,x_{5}+x_{2}\,x_{3}\,x_{4}+x_{2}\,x_{3}\,x_{5}+x_{2}\,x_{4}\,x_{5}+x_{3}\,x_{4}\,x_{5}=-{\frac {d}{a}},}
x
1
x
2
x
3
x
4
+
x
1
x
2
x
3
x
5
+
x
1
x
2
x
4
x
5
+
x
1
x
3
x
4
x
5
+
x
2
x
3
x
4
x
5
=
e
a
,
{\displaystyle x_{1}\,x_{2}\,x_{3}\,x_{4}+x_{1}\,x_{2}\,x_{3}\,x_{5}+x_{1}\,x_{2}\,x_{4}\,x_{5}+x_{1}\,x_{3}\,x_{4}\,x_{5}+x_{2}\,x_{3}\,x_{4}\,x_{5}={\frac {e}{a}},}
x
1
x
2
x
3
x
4
x
5
=
−
f
a
.
{\displaystyle x_{1}\,x_{2}\,x_{3}\,x_{4}\,x_{5}=-{\frac {f}{a}}.}
Точной формулы решения уравнения пятой степени в радикалах не существует . Если
a
=
1
,
f
=
0
{\displaystyle a=1,f=0}
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
=
0
{\displaystyle x^{5}+bx^{4}+cx^{3}+dx^{2}+ex=0}
x
{\displaystyle x}
Сводное уравнение )
x
(
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
)
=
0
{\displaystyle x(x^{4}+bx^{3}+cx^{2}+dx+e)=0}
корней равен нулю .
В скобках уравнение четвертой степени .
Если
b
=
d
=
0
{\displaystyle b=d=0}
биквадратное . Один из корней равен нулю, остальные корни ищут по формуле
±
−
c
±
c
2
−
4
e
2
{\textstyle \pm {\sqrt {-c\pm {\frac {\sqrt {c^{2}-4e}}{2}}}}}
Если
b
=
e
=
0
{\displaystyle b=e=0}
x
4
+
c
x
2
+
d
x
=
0
{\displaystyle x^{4}+cx^{2}+dx=0}
x
(
x
3
+
c
x
+
d
)
=
0
{\displaystyle x(x^{3}+cx+d)=0}
формуле Кардано .
Решите уравнение
x
5
+
5
x
=
0
{\displaystyle x^{5}+5x=0}
Решение. Выносим
x
{\displaystyle x}
x
(
x
4
+
5
)
=
0
{\displaystyle x(x^{4}+5)=0}
Раскладываем
x
4
+
5
{\displaystyle x^{4}+5}
x
(
x
−
5
4
2
−
5
4
2
i
)
(
x
+
5
4
2
−
5
4
2
i
)
(
x
+
5
4
2
+
5
4
2
i
)
(
x
−
5
4
2
+
5
4
2
i
)
=
0
{\displaystyle x(x-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)(x+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)(x+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)(x-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)=0}
Уравнение имеет пять корней:
x
1
=
0
{\displaystyle x_{1}=0}
x
2
=
5
4
2
+
5
4
2
i
{\displaystyle x_{2}={\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}
x
3
=
−
5
4
2
+
5
4
2
i
{\displaystyle x_{3}=-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}
x
4
=
−
5
4
2
−
5
4
2
i
{\displaystyle x_{4}=-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}
x
5
=
5
4
2
−
5
4
2
i
{\displaystyle x_{5}={\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}
![{\displaystyle x(x-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)(x+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)(x+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)(x-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05125aff544f16172944f2367313b7a04c419597)
![{\displaystyle x_{2}={\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8141ffebc487d6eb18fda065b7a4c36f03eea5f3)
![{\displaystyle x_{3}=-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}+{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7152304a4e45b1fcd5ce898e6f15a79f00a9ef8)
![{\displaystyle x_{4}=-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cafca32927c86a14089677fde13305bb63ebc4c1)
![{\displaystyle x_{5}={\frac {\sqrt[{4}]{5}}{\sqrt {2}}}-{\frac {\sqrt[{4}]{5}}{\sqrt {2}}}i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb434032f0921bfc61c282236dd96a3e1b2ff01)