The polynomial x2+5x+3 in R[x] induces the function f:RR defined
by the rule: f(a) = a2+5a+3 for each a in R.  This assignment defines an surjective ring homomorphism R[x] R[a] = ring of polynomials in the variable a.

Remark:  There is a distinction between polynomials and polynomial functions (these rings are not isomorphic in general).

Example:  Consider the unequal polynomials x4+x+1 and x3+x2+1 in
Z3[x] and the induced functions
 f: Z3Z3 , f(a)= a4+a+1 for each a in Z3
 g: Z3Z3, g(a)=a3+a2+1 for each a in Z3

 f(0)=1=g(0); f(1)=3=0, g(1)=3=0; f(2)=19=1, g(2)=13=1

Therefore, f=g, but the polynomials that induced these functions are not equal, (i.e., the above homomorphism is not always injective).

Note:  The existence of such an example is clear, since the polynomial ring Z3[x] is infinite, but there are only a finite number of functions f: Z3Z3.
Question:  Is there a similar example if the coefficient field is infinite?
Answer: NO!!!   R[x] isomorphic to R[a] iff R is infinite.
A Polynomial Subtlety