Question : If x + y + z = 8, and x2 + y2 + z2 = 20, then the value of x3 + y3 + z3 – 3xyz is _______.
Option 1: 16
Option 2: 10
Option 3: 15
Option 4: –16
Correct Answer: –16
Solution :
(x + y + z)
2
= x
2
+ y
2
+ z
2
+ 2xy + 2yz + 2xz
⇒ 2(xy + yz + zx) = (x + y + z)
2
– (x
2
+ y
2
+ z
2
)
⇒ 2(xy + yz + zx) = 8
2
– 20 = 64 – 20 = 44
$\therefore$ xy + yz + zx = 22
Now, x
3
+ y
3
+ z
3
– 3xyz
= (x + y + z)(x
2
+ y
2
+ z
2
– xy – yz – xz)
= 8(20 – 22)
= –16
Hence, the correct answer is –16.
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