Problem : Let be polynomials in , each having all integer coefficients, such that . Assume that is not the zero polynomial. Show that and
As are integer coefficient polynomials so gives integer values at integer points.
Now as is not zero polynomial
Then or as =integer
But it is given that
This implies . This is only possible if
Hence the values of x for which is non-zero, are all zero. The values of x for which , we have implying each is zero.
Finally implies . Since hence it is 1.
- Topic: Theory of Equations
- Central idea: Sum of square quantities zero implies each of the quantities is zero.
- Course: I.S.I. & C.M.I. Entrance Program