Re: [中學] 一題代數
※ 引述《Herseth (Memento)》之銘言：
: x + y + z=0
: x^3 + y^3 + z^3 = 3
: x^5 + y^5 + y^5 = 15
: 試求 x^2 + y^2 + z^2 = ?
: 感謝高手解惑
提供另一作法
x^3 + y^3 + z^3 -3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx) = 0
所以 xyz = 1
z = -(x+y), 所以 xy(x+y) = -1
x^5 + y^5 + z^5 = (x+y)^5 + z^5 - 5xy(x^3+y^3) - 10x^2y^2(x+y)
= (x+y+z)(...) -5xy(x+y)(x^2-xy+y^2) - 10xyxy(x+y)
= 0 + 5(x^2-xy+y^2) + 10xy
= 5(x^2+xy+y^2)
所以 x^2+xy+y^2=3
x^2+y^2+z^2 = x^2 + y^2 + (x+y)^2 = 2(x^2+xy+y^2) = 6
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◆ From: 220.132.177.99
※ 編輯: dogy007 來自: 220.132.177.99 (11/24 18:58)
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11/24 22:04, , 1F
11/24 22:04, 1F
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