I have a question about the following definition from my book:

Quote Originally Posted by definition
Let V be a module over a commutative ring A. A function Q: V -> A is called a quadratic form on V if the following properties are satisfied:
(1) Q(ax) = a^2Q(x) for all a\in A and x\in V
(2) (x,y) -> Q(x+y) -Q(x)-Q(y) is a bilinear form.
My questions are the following:
  • When I looked up the definition of a quadratic form over a field property I saw that (1) is sufficient. Why should property (2) in this case also be included in the definition? Can anyone give an example of a function that is not a quadratic form, but for which (1) holds?