Concept:

Nature of roots of a cubic polynomial:

• If the maximum and minimum are of opposite signs, the cubic has three real roots.
• If one of them is zero, two of the three roots are equal.
• If both of them are zero, all three roots are equal.
• If the maximum and minimum are of the same sign, the cubic has one real and two imaginary roots.

Calculations:

Given equation is (x - a)³ + (x - b)³ + (x - c)³ = 0

Consider, f(x) = (x - a)³ + (x - b)³ + (x - c)³

Differentiating both sides, we get

f'(x) = 3(x - a)² + 3(x - b)² + 3(x - c)²

f'(x) > 0

Hence, f is an increasing function.

To find the maximum and minimum of the function, put x = ∞ in f(x):

Now, f(∞) = ∞

f(-∞) = ∞

Here, the maximum and minimum of the function are of the same sign.

Hence, the equation (x - a)³ + (x - b)³ + (x - c)³ = 0 has one real and two imaginary roots.