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

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Solution

a, b, c and p are rational. p is not a perfect cube.
so p¹/³ and p²/³ are not rational numbers. Then p is not 1 or 0.
p²/³ = p¹/³ * p¹/³ and so they are not equal.

LHS = a + b p¹/³ + c p²/³ = 0 --- (1) Given p is not a perfect cube. p is not 0 or 1. Also are irrational. Multiply (1) by p^1/3 to get: Substitute the value of c in (1) to get: So p^1/3 is imaginary. It is a contradiction as p is a rational number. Given quadratic isn't valid. So a = b = c = 0.

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