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 in the form of a recursive formula
According to mathematical induction we can examine any n-degree polynomial function using shown method.
Therefore, the polynomial   f (x) =  yanxn + an-1xn-1 + an-2xn-2 + . . .  + a2x2 + a1x + a0
we can write as
while,  for   k = 0,            an = an,
and from an-k, for  k = n,            a0 = f (x0) = y0.

Thus, expanded form of the above sum is
y - y0 = an(x - x0)n + an-2(x - x0)n-2 + . . .  + 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  + . . .  a2x2 + a1x
in the direction of the x-axis and the y-axis of a Cartesian coordinate system.
Therefore, every polynomial written in the general form can be transformed into translatable form by calculating the coordinates of translations x0 and y and the coefficients a of its source function.
Note that the complete source polynomial has n - 1 terms, missing second and the absolute term.
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
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
College algebra contents C