If x + y = 4, xy = 2, y + z = 5, yz = 3, z + x = 6 and zx = 4, then find the value of $x^3 + y^3 +z^3 - 3xy.$
Answer & explanation
Correct answer: option 4
If x + y = 4, xy = 2, y + z = 5, yz = 3, z + x = 6 and zx = 4
We know that,
x3 + y3 + z3 - 3xyz = (x + y + z) × (x2 + y2 + z2 – xy – yz – zx)
(x + y)2 = x2 + y2 + 2xy
x + y = 4, y + z = 5, z + x = 6
x + y + z = 7.5
xy + yz + zx = 2 + 3 + 4 = 9
Now,
(x + y)2 = x2 + y2 + 2xy
x2 + y2 = 16 – 4 = 12 Because..(2xy = 4)
(y + z)2 = y2 + z2 + 2yz
y2 + z2 = 25 – 6 = 19 Because...(2yz = 6)
(x + z)2 = x2 + z2 + 2xz
x2 + z2 = 36 – 8 = 28 Because...(2zx = 8)
So,
2x2 + 2y2 + 2z2 = 12 + 19 + 28 = 59
x2 + y2 + z2 = 29.5
Now,
x3 + y3 + z3 - 3xyz = (x + y + z) × (x2 + y2 + z2 – xy – yz – zx)
= (7.5) × (29.5 – 9)
= 7.5 × 20.5
= 153.75