If I am interpreting you correctly, you are wondering how you can say that if two polynomials over agree for all values of then they must be equal. In other words, if two polynomials induce the same function they are the same polynomial. This is true more generally over any infinite integral domain . First prove it's true for an infinite field by noting first that this problem is evidently equivalent to showing that a non-zero polynomial may not be simultaneously zero, which is clear since if is such with then we know that , and so in particular cannot vanish for every value of (since is infinite). For the more general case merely show that if a non-zero polynomial over an integral domain induced the zero function then it would induce the zero function over[ , and apply the previous result.

What does this mean?and how can you replace this expression-

F(x)=A(x-alpha1)(x-alpha2)....)x-alphar)

with F(x)=A(x-alpha1)^m1(x-alpha2)^m2....(x-alphar)^mr, with A=an

mk are integers >=1 and are multiplicities of the roots mk being the multiplicity of alphak

thanks very much!