where n is a natural number, the coefficients are elements of R, Here X is a formal symbol, whose powers Xi are at this point just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence (an,…,a1,a0). Two polynomials sharing the same value of n are considered to be equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term aiXi is interpreted as a polynomial that has zero coefficients at all powers of X other than Xi. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule