Note. Please provide only a hint along with some explanation, but not the answer. I want to struggle with this problem. This is not homework.
Show that for any number $c$, a polynomial $ P(x) = b_0 + b_1 x + b_2 x^2 + \cdots b_n x^n$ can also be written as $P(x) = a_0 + a_1 (x - c) + a_2 ( x- c)^2 + a_3(x - c)^3\cdots a_n(x - c)^n$ where $a_0 = P(c)$.