
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 n^{th}
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 n^{th}
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 a_{n},
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 yaxis, xaxis
and both axes, respectively. 
Observe that the
first criteria of the classification separates even and odd n^{th}
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
 y_{0}
= a_{n}(x
 x_{0})^{n}, 
where x_{0}
= 
a_{n}_{
}_{}_{
}_{1}/(na_{n}),
y_{0}
=
f
(x_{0})
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
(x_{0},
y_{0}) 
and
the real roots if a_{n}
y_{0 }
<
0, 

as
shows the right figure (where a_{n}
> 0 ). 



For
n
= 2m + 1, m Î
N the odd power function 
has,
the point of inflection I
(x_{0},
y_{0}),
a_{t}
=
0 
and
the root 

as
shows the right figure (where a_{n}
> 0). 








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