If x, y, z are real numbers such that x + y + z = 10 and xy + yz + zx = 18, then what is the value of x³ + y³ + z³ - 3xyz?
460
We know that x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx) x + y + z = 10 xy + yz + zx = 18 (x + y + z)² = x² + y² + z² + 2(xy + yz + zx) 10² = x² + y² + z² + 2(18) 100 = x² + y² + z² + 36 x² + y² + z² = 64 x³ + y³ + z³ - 3xyz = (10)(64 - 18) = 10(46) = 460
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