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det (a
ij
) 6= 0 a
0j
= δ
0j
x =
n−1
P
i=0
γ
i
y
i
=
n−1
P
j=0
(
n−1
P
i=0
γ
i
a
ij
)x
j
⇒
n−1
P
i=0
a
ij
γ =
δ
1j
, j = 1, ..., n − 1 γ
i
det (a
ij
) 6= 0
f(x)
S
4
A
2
= (x
1
x
2
+ x
3
x
4
)(x
1
x
3
+ x
2
x
4
) + (x
1
x
2
+ x
3
x
4
)(x
1
x
4
+ x
2
x
3
) + (x
1
x
3
+
x
2
x
4
)(x
1
x
4
+ x
2
x
3
) = x
2
1
x
2
x
3
+ ... = σ
1
σ
3
− 4σ
4
A
3
= (x
1
x
2
+ x
3
x
4
)(x
1
x
3
+ x
2
x
4
)(x
1
x
4
+ x
2
x
3
) = x
3
1
x
2
x
3
x
4
+ ... = σ
2
1
σ
4
+
σ
2
3
− 4σ
2
σ
4
S
4
B
1
= (x
1
+ x
2
)(x
3
+ x
4
) + (x
1
+ x
3
)(x
2
+ x
4
) + (x
1
+ x
4
)(x
2
+ x
3
) = 2σ
2
B
2
= (x
1
+ x
2
)(x
3
+ x
4
)(x
1
+ x
3
)(x
2
+ x
4
) + (x
1
+ x
2
)(x
3
+ x
4
)(x
1
+ x
4
)(x
2
+
x
3
) + (x
1
+ x
3
)(x
2
+ x
4
)(x
1
+ x
4
)(x
2
+ x
3
) = x
2
1
x
2
2
+ ... = σ
2
2
+ σ
1
σ
3
− 4σ
4
B
3
= (x
1
+ x
2
)(x
1
+ x
3
)(x
1
+ x
4
)(x
2
+ x
3
)(x
2
+ x
4
)(x
3
+ x
4
) = x
3
1
x
3
2
+ ... =
σ
1
σ
2
σ
3
− σ
2
1
σ
4
+ σ
2
3
3210 σ
1
σ
2
σ
3
3111 σ
2
1
σ
4
2220 σ
2
3
2211 σ
2
σ
4
x
1
= x
2
= x
3
= 1, x
4
= 0. σ
1
= 3, σ
2
= 3, σ
3
= 1, σ
4
= 0 8 = 9 + B B =
−1
x
1
= x
2
= x
3
= x
4
= 1. σ
1
= 4, σ
2
= 6, σ
3
= 4, σ
4
= 1
64 = 96 + 16A + 16B + 6C 32 = 48 + 8A + 8B + 3C − 16 = 8A + 8B + 3C
x
1
= x
2
= 1, x
3
= x
4
= −1. σ
1
= 0, σ
2
= −2, σ
3
= 0, σ
4
= 1
0 = −2C C = 0
x
1
= x
2
= x
3
= 1, x
4
= −1. σ
1
= 2, σ
2
= 0, σ
3
= −2, σ
4
= −1
0 = −4B + 4C B = C 0 = −4A + 4B A = B = −1
, ãäå a0j = δ0j .
det (aij ) 6= 0
Äîêàçàòåëüñòâî. Èìååì x = P γi yi = P ( P γi aij )xj ⇒ P aij γ =
n−1 n−1 n−1 n−1
i=0 j=0 i=0 i=0
δ1j , j = 1, ..., n − 1. Ýòà ñèñòåìà óðàâíåíèé (îòíîñèòåëüíî γi ) èìååò åäèí-
ñòâåííîå ðåøåíèå òîãäà è òîëüêî òîãäà, êîãäà det (aij ) 6= 0.
Ïðåäëîæåíèå 21. Âñÿêèé ìíîãî÷ëåí f (x) ñòåïåíè 5 îáðàòèìûì ïðå-
îáðàçîâàíèåì ×èðíãàóçåíà ìîæåò áûòü ïðèâåäåíî ê íîðìàëüíîìó âèäó.
Äîêàçàòåëüñòâî îñíîâàíî íà íåïîñðåäñòâåííîì äîñòàòî÷íî ãðîìîçäêîì
âû÷èñëåíèè.
Ïðèëîæåíèå.
1) Ê âû÷èñëåíèþ êóáè÷åñêèõ ðåçîëüâåíò (â S4 )
A2 = (x1 x2 + x3 x4 )(x1 x3 + x2 x4 ) + (x1 x2 + x3 x4 )(x1 x4 + x2 x3 ) + (x1 x3 +
x2 x4 )(x1 x4 + x2 x3 ) = x21 x2 x3 + ... = σ1 σ3 − 4σ4
A3 = (x1 x2 + x3 x4 )(x1 x3 + x2 x4 )(x1 x4 + x2 x3 ) = x31 x2 x3 x4 + ... = σ12 σ4 +
2
σ3 − 4σ2 σ4.
2) Ê âû÷èñëåíèþ âòîðîé êóáè÷åñêîé ðåçîëüâåíòû (â ) S4
B1 = (x1 + x2 )(x3 + x4 ) + (x1 + x3 )(x2 + x4 ) + (x1 + x4 )(x2 + x3 ) = 2σ2 ,
B2 = (x1 + x2 )(x3 + x4 )(x1 + x3 )(x2 + x4 ) + (x1 + x2 )(x3 + x4 )(x1 + x4 )(x2 +
x3 ) + (x1 + x3 )(x2 + x4 )(x1 + x4 )(x2 + x3 ) = x21 x22 + ... = σ22 + σ1 σ3 − 4σ4 ,
B3 = (x1 + x2 )(x1 + x3 )(x1 + x4 )(x2 + x3 )(x2 + x4 )(x3 + x4 ) = x31 x32 + ... =
.
σ1 σ2 σ3 − σ12 σ4 + σ32
3210 σ1 σ2 σ3
3111 σ12 σ4
2220 σ32
2211 σ2 σ4
(
x1 = x2 = x3 = 1, x4 = 0. σ1 = 3, σ2 = 3, σ3 = 1, σ4 = 0 8 = 9 + B B =
−1 )
x1 = x2 = x3 = x4 = 1. σ1 = 4, σ2 = 6, σ3 = 4, σ4 = 1
64 = 96 + 16A + 16B + 6C 32 = 48 + 8A + 8B + 3C − 16 = 8A + 8B + 3C
x1 = x2 = 1, x3 = x4 = −1. σ1 = 0, σ2 = −2, σ3 = 0, σ4 = 1
(0 = −2C C = 0 )
x1 = x2 = x3 = 1, x4 = −1. σ1 = 2, σ2 = 0, σ3 = −2, σ4 = −1
(0 = −4B + 4C B = C 0 = −4A + 4B A = B = −1 )
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