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Question -

If x + y + z = 8 and xy + yz+ zx = 20, find the value of x3 + y3 + z3 – 3xyz.



Answer -

x3 + y3 + z3 –3xyz = (x + y + z) (x2 + y2 + z2 -xy-yz – zx)
Now, x + y + z = 8
Squaring, we get
(x + y + z)2 = (8)2
x2 + y2 + z2 + 2(xy + yz+ zx) = 64
x2 +y2 + z2 + 2 x 20 = 64
 x2 + y2 + z2 + 40 = 64
 x2 + y2 + z2 = 64 – 40 = 24
Now,
x3 + y3 + z3 – 3xyz = (x + y +z) [x2 + y2 + z2 – (xy + yz +zx)]
= 8(24 – 20) = 8 x 4 = 32

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