The equation `(x-a)^3 + (x-b)^3 + (x-c)^3 = 0` has :

Options

A. all the roots real
B. one real and two imaginary roots
C. three real roots namely x = a, x = b, x = c
D. none of these

Answer

B

Explanation

Let `f(x) = (x-a)^(3) + (x-b)^(3) + (x-c)^(3)`. Then, `f'(x) = 3{(x-a)^(2)+(c-b)^(2)+(x-c)^(2)}`
Clearly, `f'(x) gt 0` for all x. So, f'(x) = 0 has no real roots Consequently, f(x) cannot have two distinct real roots.
Hence, f(x) = 0 has two imaginary and one real root.