Question

is the sum of two polynomials of degree 5 always a polynomial of degree 5? Give an example to explain your answer

Answer 1

A counter example is

f(x)=5x^5+2x^3+3x
g(x)=-5x^5-x^4+x^2-4

Then f(x)+g(x) = -x^4+2x^3+x^2+3x-4  which is a polynomial of degree 4.

So the answer is no.  Counter-example is given above.