推 wjx0305 :這個比較容易看懂說：） 08/18 10:01
※ 引述《wjx0305 (胖包子~)》之銘言：
: x+y+z=1
: x^2 +y^2 +z^2 =2
: x^3 +y^3 +z^3 =3
: x^4 +y^4 +z^4 =？
: 感謝：）
(x + y + z)^2 = (x^2 +y^2 +z^2) + 2(xy + yz + zx)
=> 1 = 2 + 2(xy + yz + zx)
=> 2(xy + yz + zx) = -1
(x + y + z)[x^2 +y^2 +z^2 - (xy + yz + zx) ] = x^3 + y^3 + z^3 - 3xyz
=> 1 [2 + 1/2] = 3 - 3xyz
=> xyz = 1/6
(x^3 + y^3 + z^3)(x + y + z) = (x^4 + y^4 + z^4) + xy(y^2 + x^2) +
xz(z^2 + x^2) + yz(z^2 + y^2)
=> 3 * 1 = (x^4 + y^4 + z^4) + xy(2 - z^2) + xz(2 - y^2) + yz(2 - x^2)
= (x^4 + y^4 + z^4) + -1 - (1/6)*1
=> x^4 + y^4 + z^4 = 1 + 1/6 + 3 = 25/6
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