Re: [中學] 對稱多項式

[mack (腦海裡依然記得妳)]

※ 引述《Ericdion ( 由心出發 )》之銘言：
: 兩題，第296與309
: 想了幾天還是沒找到方法
: 請高手指教，感恩！！！
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309.
  (x + y + z) * (x^2 + y^2 + z^2)
= (x^3 + y^3 + z^3) + [x^2(y + z) + y^2(z + x) + z^2(x + y)]
= (x^3 + y^3 + z^3) - 24

  (x + y + z)^2
= (x^2 + y^2 + z^2) + 2(xy + yz + zx)
=> 9 = (x^2 + y^2 + z^2) + 2(xy + yz + zx)

1/x + 1/y + 1/z = (xy + yz + zx)/(xyz) = -1/3
x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)
                       = (x^3 + y^3 + z^3) - 24 + (x + y + z)*(- xy - yz - zx)
=> 9(xy + yz + zx) = - 24 + (-3)*(- xy - yz - zx)
=> xy + yz + zx = -4

=> 9 = (x^2 + y^2 + z^2) + 2(xy + yz + zx)
     = (x^2 + y^2 + z^2) - 8
=> x^2 + y^2 + z^2 = 17

=> (x^3 + y^3 + z^3) - 24 = (x + y + z) * (x^2 + y^2 + z^2)
                          = -3 * 17
                          = -51
=> x^3 + y^3 + z^3 = -27

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