Polynomial and/or Polynomial Functions and Equations
      Sigma notation of the polynomial
         Coefficients of the source polynomial in the form of a recursive formula
         Coefficients of the source polynomial function are related to its derivative at x0
Sigma notation of the polynomial
Coefficients of the source polynomial shown by a recursive formula
Therefore, the polynomial   f (x) =  yanxn + an-1xn-1 + an-2xn-2 + ¼  + a2x2 + a1x + a0
we can write as
                        while,  for   k = 0,             an = an,
            and from an- k, for  k = n,            a0 = f (x0) = y0.
 
The expanded form of the above sum is
y - y0 = an(x - x0)n + an-2(x - x0)n-2 + ¼  + a2(x - x0)2 + a1(x - x0)
where x0 and y0 are coordinates of translations of the graph of the source polynomial
fs(x) = anxn + an-2xn-2 + ¼  + a2x2 + a1x
in the direction of the x-axis and the y-axis of a Cartesian coordinate system.
Let's finally mention the main property of a polynomial that shows the nature of the revealed theory the best. 
An n-th degree polynomial function and all its successive derivatives to the (n - 1)-th order, have constant horizontal translation x0.
Coefficients of the source polynomial function are related to its derivative at x0
The coefficients of the source polynomial are related to corresponding value of its derivative at x0 like the coefficients of the Taylor polynomial in Taylor's or Maclaurin's formula, thus
since,  an = anan-1 = 0a0 =  f (x0), and where  f (n - k)(x0) denotes the (n - k)th derivative at x0.
Such for example, the coefficient a1 of the source cubic of   f(x) = a3x3 + a2x2 + a1x + a0 
since   f ' (x) = 3a3x2 + 2a2x + a1   and  x0 = - a2/(3a3)  then 
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