If a, b, c are real and x3−3b2x+2c3 is divisible by x−a and x−b, then
As f(x)=x3−3b2x+2c3 is divisible by x−a and x−b, therefore
f(a)=0 ⇒ a3−3b2a+2c3=0 ......(i)
and f(b) = 0 ⇒ b3−3b3+2c3=0 .... (ii)
From (ii), b = c
From (i), a3−3ab2+2b3=0 (putting b = c)
⇒ (a−b)(a2+ab−2b2)=0
⇒ a = b = c or a2+ab=2b2 and b = c
∴ a2+ab−2b2
and b = c is equivalent to a= -2b = -2c