Question

Let a, b , c be rational numbers such that p is not a perfect cube and a + bp 1/3 + cp 2/3 = 0 . prove that a=b=c

Answer 1

    a, b, c and p are rational.  p is not a perfect cube.  so p¹/³ and p²/³ are not rational numbers. Then if p is not 1, p²/³ = p¹/³ * p¹/³   and so they are not equal too.

         LHS = a + b p¹/³ + c p²/³ = 0       --- (1)
           =>  b p¹/³ + c p²/³ = - a                  (2)

Cube on both sides.
=> b³ p+c³ p² + 3 b² c p⁴/³ + 3 b c² p⁵/³ = - a³
=> (a³ + b³ p + c³ p²) + (3 b²cp) p¹/³+ (3 bc²p) p²/³ = 0   --- (3)

compare (1) and (3): then:  The ratios of coefficients are equal.
       (a³+b³p + c³p²) / a  = 3 b²cp/ b = 3 bc² p /c
                                     = 3 b c p
       a³ + b³ p + c³ p² - 3 a b c p = 0
       a³ +  (b p¹/³)³ + (c p²/³)³ - 3 a (b p¹/³) (c p²/³) = 0

Hence, from the formula we know, we get
            a + b p¹/³ + c p²/³ = 0