If x, y, z satisfy:
x + y + z = 12,
1/x + 1/y + 1/z = 2, and
x3 + y3 + z3 = -480,
find x2y + xy2 + x2z + xz2 + y2z + yz2.
Tag the three equations by E1, E2, E3, and the expression to be evaluated by EXP.
EXP factors and = xy(x + y) + xz(x + z) + yz(y + z).
E1 gives values for (x + y), (x + z), and (y + z). Plug those into EXP.
EXP = xy(12 - z) + xz(12 - y) + yz(12 - x) = 12(xy + xz + yz) - 3xyz
Clear the denominators in E2.
xy + xz + yz = 2xyz
Substituted back gives EXP = 21xyz.
So we need values for x,y,and z or more likely, the product xyz.
That's all for now. When I have more time I'll consider E3.
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Posted by xdog
on 2025-03-29 16:37:01 |
a start
