Is the difference of two polynomials P1 and P2 always a polynomial?
1 Answer
Yes
Explanation:
If you add, subtract or multiply any two polynomials then the result will be a polynomial.
When adding or subtracting two polynomials you typically group together similar terms and add or subtract their coefficients.
The addition, subtraction and multiplication of polynomials in general has the following properties:

There is a polynomial which acts as an additive identity, namely the zero polynomial
#0# . For any polynomial#P# :#P + 0 = 0 + P = P# 
Any polynomial has an additive inverse formed by reversing the signs on all of its coefficients.

Addition of polynomials is commutative and associative.

There is a polynomial which acts as a multiplicative identity, namely the polynomial
#1# . 
Multiplication of polynomials is (commutative and) associative.

Multiplication is left and right distributive over addition. That is, for any three polynomials,
#P, Q# and#R# we have:#P(Q+R) = PQ+PR# and#(P+Q)R = PR+QR#
These properties loosely mean that, regardless of whether you are dealing with polynomials with integer, rational, Real or Complex coefficients and regardless of whether they are in one or more variables, the set of all such polynomials forms an algebraic structure called a ring and usually a commutative ring.