Examples

• The polynomial function associated with X² - 2X + 2 in R[X] is the (quadratic) function R -> R that sends, for example, 3 to 3² - 2 · 3 + 2 = 5. Its graph is a parabola. By completing the square
  X² - 2X + 2 = (X - 1)² + 1,

  it follows easily that the polynomial function has no zeros, since (r - 1)² + 1 is positive for every real number r.

• Put p(X) = X³ - X Z/3Z[X]. The corresponding polynomial function is the zero function! Every element of Z/3Z is a zero of p(X). This illustrates that in general the polynomial function does not determine the polynomial. This is a good reason by itself for distinguishing a polynomial and the corresponding polynomial function.
• Finding the zeros of a polynomial is nontrivial in general. For polynomials of degree 1 or 2, the usual high school methods often work. For example, to find the zeros of X² + 11X + 4 in Z/13Z[X], we first complete the square, using 11 = 2 · (-1) mod 11,
  X² + 11X + 4 = (X - 1)² + 3.

  Then we find all elements in Z/13Z whose square is -3. There are two such numbers: 6² = (-6)² = -3, so that the zeros are

  1 + 6 = 7 and 1 - 6 = -5.