一道数学技巧题X1 + X2 + X3 + X4 = aX1^2 + X2^2 + X3^2 + X4^2 = bX1^
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一道数学技巧题
X1 + X2 + X3 + X4 = a
X1^2 + X2^2 + X3^2 + X4^2 = b
X1^3 + X2^3 + X3^3 + X4^3 = c
X1^4 + X2^4 + X3^4 + X4^4 = d
求X1 * X2 * X3 * X4
a,b,c,d已知
X1 + X2 + X3 + X4 = a
X1^2 + X2^2 + X3^2 + X4^2 = b
X1^3 + X2^3 + X3^3 + X4^3 = c
X1^4 + X2^4 + X3^4 + X4^4 = d
求X1 * X2 * X3 * X4
a,b,c,d已知
对称多项式,等价于
(X1)=a
(X1^2)=b
(X1^3)=c
(X1^4)=d
so,
2(X1X2)=(X1)(X1)-(X1^2)
3(X1X2X3)=(X1X2)(X1)-(X1)(X1^2)+(X1^3)
4(X1X2X3X4)=(X1X2X3)(X1)-(X1X2)(X1^2)+(X1)(X1^3)-(X1^4)
X1X2X3X4=(X1X2X3X4)
=1/24(a^4-6a^2*b+3*b^2+8ac-6d)
(X1)=a
(X1^2)=b
(X1^3)=c
(X1^4)=d
so,
2(X1X2)=(X1)(X1)-(X1^2)
3(X1X2X3)=(X1X2)(X1)-(X1)(X1^2)+(X1^3)
4(X1X2X3X4)=(X1X2X3)(X1)-(X1X2)(X1^2)+(X1)(X1^3)-(X1^4)
X1X2X3X4=(X1X2X3X4)
=1/24(a^4-6a^2*b+3*b^2+8ac-6d)
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