A "non-polynomial" can be just about anything; how alike they are depends what function (or non-function) you specifically have in mind.
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Adding and subtracting polynomials is simply the adding and subtracting of their like terms.
The sum of two polynomials is always a polynomial. Therefore, it follows that the sum of more than two polynomials is also a polynomial.
division
No. Even if the answer is zero, zero is still a polynomial.