Sum of polynomials are always polynomials.
Note that despite it's name, single constants, monomials, binomials, trinomials, and expressions with more than three terms are all polynomials.
For example,
0, π sqrt(2)x, 4x+2, x^2+3x+4, x^2-x^2, x^5+x/ π -1
are all polynomials.
What makes an expression NOT a polynomial?
Expressions that contain non-integer or negative powers of variables, rational functions, infinite series.
For example,
sqrt(x+1), 1/x+4, 1+x+ x^2/2!+x^3/3!+x^4/4!+...., (5x+3)/(6x+7)
are NOT polynomials.