Hello, I am puzzled about this operation (1) with polynomials in a general proof of Exponential input theorem and I don't understand the the concept of a being s-fold zero.
p(D) = q(D)(D - a)s
q(a) = nonzero
The above implies
p(s)(a) = q(a)s!
The proof lets k be the degree of q(D) and writes it in powers of (D-a)
(1) q(D) = q(a) + c1(D - a) + ... ck(D - a)k
(2) q(D) = q(a)(D−a)s+c1(D−a)s+1 +...+ck(D−a)s+k
then substitute D with a
How did they figure (1) out?
if k = 2, I get
D2 + c1D +c2 = A2 + c1A + c2 + c1D - c1A + c2D2 - c22DA + c2A2
I feel totally confused and as if I missed a lot
Could you please point out what I do not understand?