A revealing insight into the polynomial function - continue
Benefits the principle provide
Let's show here only a few of the benefits the principle brings.
Classification of the polynomial functions
The first, the principle enables classification of the polynomial functions meaning, it provides basic conditions for particular type of function, or group of functions, an nth degree polynomial includes.
That is, defines the form of the graph specific type of a polynomial function is represented.
Thus for example, cubic polynomial consists of three types, or is represented by three distinct forms (shapes) of graphs of functions.
Using the same approach classified is the quartic polynomial to ten types of characteristic functions and so on.
Each nth degree polynomial is classified using the same basic criteria that include, examination of appearance or exception of each of the source coefficients a, and depending on the sign of their product with the leading coefficient an, since the source coefficients show the direction of the vertical translation of the corresponding derivative.
The classification also defines all variations of a polynomial expression related to the changes of polarity of the variables such as,   f (-x),   - f (x)  and   - f (-x)  that cause the reflections of the graph of the polynomial function around the  y-axis, x-axis and both axes, respectively.
Observe that the first criteria of the classification separates even and odd nth degree polynomials called the power functions or monomials as the first type, since all coefficients a of their source function vanished, (see the above diagram).
Therefore, we write the translated power (or monomial) function or the first type
y - y0 = an(x - x0)n,
where  x0 = - an - 1/(nan),   y0 = f (x0) and n is an even or an odd positive integer.
For  n = 2m, m Î N the even power function has,
  the turning point  T (x0, y0)
  and the real roots if   an y0  < 0,
 
as shows the right figure (where an > 0 ).
For  n = 2m + 1, m Î N the odd power function
  has, the point of inflection I (x0, y0),  at = 0
  and the root
 
as shows the right figure (where an > 0).
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