Definition of $f+g$ and $f \cdot g$ in the context of polynomial rings? I've been asked to prove that given $f$ and $g$ are polynomials in $R[x]$, where $R$ is a commutative ring with identity, 
$(f+g)(k) = f(k) + g(k)$, and $(f \cdot g)(k) = f(k) \cdot g(k)$. However, I always took these things as definition. What exactly is there to prove? How exactly are $f+g$ and $f \cdot g$ defined here?

One construction (I do not like using constructions as definitions) of $R[x]$ is as the set of formal sums $\sum r_i x^i$ of the symbols $1, x, x^2, x^3, ...$ with coefficients in $R$. Addition here is defined pointwise and multiplication is defined using the Cauchy product rule. For every $k \in R$ there is an evaluation map $\text{eval}_k : R[x] \to R$ which sends $\sum r_i x^i$ to $\sum r_i k^i$, and the question is asking you to show that this map is a ring homomorphism. 
To emphasize that this doesn't follow immediately from the construction, note that this statement is false if $R$ is noncommutative. 
The reason I don't want to use the word "definition" above is that "polynomial ring" should more or less mean "the ring such that the above thing is true, and if it isn't, we messed up and need a different construction." 
