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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