Re: [中學] 多項式根問題
※ 引述《jimlucky (......)》之銘言：
: a, b, c 是方程式2x^3-x^2+3x-1=0的根
: 求1/a^3+1/b^3+1/c^3
: 嘗試用三根和、兩兩乘積和、三根積去換，換不太出來~@@
: 想請教各位大大了，感謝~
3 2
f(1/x)= -x + 3x - x + 2 = 0
由根與係數可知
1/a + 1/b + 1/c = 3
1/ab + 1/bc + 1/ca = 1
1/abc = 2
因此 (1/a)^3 + (1/b)^3 + (1/c)^3
= (1/a + 1/b + 1/c) * ( (1/a)^2 + (1/b)^2 + (1/c)^2 - 1/ab - 1/bc - 1/ca )
+ 3/abc
= 3 * ( 7 - 1 )
+ 3 * 2
= 24
中間會用到 x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+zx)
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噢噢 還好你沒被我誤導...用鍵盤算果然很容易錯 T_T
也謝謝你糾正我！
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