Try this beautiful problem from TOMATO Subjective Problem no. 173 based on the Sum of Polynomials.

* Problem* : Sum of polynomials

Let be polynomials in , each having all integer coefficients, such that . Assume that is not the zero polynomial. Show that and

**Solution*** *:

As are integer coefficient polynomials so gives integer values at integer points.

Now as is not zero polynomial

for some

Then or as =integer

or

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.

Therefore .

Finally implies . Since hence it is 1.

(Proved)

*Chatushpathi*

**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- Video: Primes and Polynomials