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Question
Solve for x, y and z if: x + y + z = 1 ; x^2 + y^2 +z^2 = 35 and x^3 + y^3 + z^3 = 97. Thanks.

Answer
x + y + z = 1

(x + y + z)^2 = 1

x^2 + y^2 +z^2 + 2xy + 2xz + 2yz = 1

35 + 2xy +2xz + 2yz = 1

2xy +2xz + 2yz = -34

xy + xz + yz = -17


-17 = (1)(xy + xz + yz ) = (x + y + z)(xy + xz + yz )

-17 = 3xyz + yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2


(x + y + z)^3 = 1

x^3 + y^3 + z^3 + 3(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 6xyz  = 1

97 + 3(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 6xyz  = 1
 
3(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 6xyz  = -96

(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 2xyz  = -32

Subtract the following two equations

-17 = 3xyz + yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2
-32 = 2xyz + yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2


15 = xyz


We now have

x + y + z = 1

xy + xz + yz = -17

xyz = 15


So,

(t-x)(t-y)(t-z) = t^3 - t^2 - 17t - 15


The roots of t^3 - t^2 - 17t - 15 are 5 , -1 , -3

So any permutation of 5 , -1 , -3 may be assigned to x , y , z

This gives 6 possible solutions to the given 3 equations

x=5 y=-1 z=-3 ; x=5 y=-3 z=-1 ; x=-1 y=5 z=-3 ; x=-1 y=-3 z=5 ; x=-3 y=-1 z=5 ;x=-3 y=5 z=-1  

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