x,y,z满足x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求x,y,z,x^4+y^4+z^4等于几

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x,y,z满足x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求x,y,z,x^4+y^4+z^4等于几
x,y,z满足x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求x,y,z,x^4+y^4+z^4等于几

x,y,z满足x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求x,y,z,x^4+y^4+z^4等于几
x+y+z=3 （1）
x²+y²+z²=29 （2）
x³+y³+z³=45 （3）
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由方程（1）得
x+y=3-z （4）
(x+y)²=(3-z)²
x²+y²+2xy=9-6z+z²
x²+y²+z²+2xy=9-6z+2z²
29+2xy=9-6z+2z²
xy=z²-3z-10 （5）
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由方程（3）得
(x+y)(x²-xy+y²)+z³=45
代入（4）和（5）得
(3-z)(29-z²-z²+3z+10)+z³=45
(3-z)(39-2z²+3z)+z³=45
z³-3z²-10z+24=0
(z³-6z²+12z-8)+(3z²-12z+12)-10z+20=0
(z-2)³+3(z-2)²-10(z-2)=0
(z-2)[(z-2)²+3(z-2)-10]=0
(z-2)(z-2+5)(z-2-2)=0
(z-2)(z+3)(z-4)=0
z=2,-3,4
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z=2时,xy=-12,x+y=1 解得 x=4,y=-3 或 x=-3,y=4
z=-3时,xy=8,x+y=6 解得 x=4,y=2 或 x=2,y=4
z=4时,xy=-6,x+y=-1 解得 x=2,y=-3 或 x=-3,y=2
原方程组的解是
x = 4,y = 2,z = -3
x = -3,y = 2,z = 4
x = 2,y = 4,z = -3
x = -3,y = 4,z = 2
x = 4,y = -3,z = 2
x = 2,y = -3,z = 4
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x^4+y^4+z^4
=(x²+y²)²-2x²y²+z^4
=(29-z²)²-2(z²-3z-10)²+z^4
=12z³-36z²-120z+641
=12(z-2)(z-4)(z+3)+353
=353

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