


Let's
set up the missing
cornerstone! 

A
revealing insight into the polynomial function 
Every
polynomial function has its initial position at the origin of the
coordinate system 
There
are three methods to transform polynomial expression 
f
(x)
= a_{n}x^{n}
+ a_{n
}_{}_{
}_{1
}x^{n}^{
}^{}^{
}^{1}
+ a_{n}_{
}_{}_{
}_{2}_{
}x^{n}^{
}^{
}^{2}
+
. . . +
a_{2}x^{2}
+
a_{1}x + a_{0} 
into
its source form 
f_{s}(x)
= a_{n}x^{n}
+ a_{n
}_{}_{
}_{2}_{
}x^{n}^{
}^{}^{
}^{2}
+ a_{n
}_{}_{
}_{3
}x^{n}^{
}^{}^{
}^{3}
+
. . . +
a_{3}x^{3}
+
a_{2}x^{2}
+ a_{1}x, 
whose
graph passes through the origin. 
Each
method is based on the fact that a polynomial written in
general form represents translation of its source
(original) function in the direction of the
coordinate axes, where the coordinates of translations
are 

Therefore,
each polynomial missing second term (a_{n
}_{}_{
}_{1}
=
0),
represents a source polynomial function whose graph is
translated in the direction of the yaxis
by
y_{0}
= a_{0}. 
Let's
reveal all three methods. 

First
method 
If
we plug the coordinates of translations with changed
signs into a given polynomial y
=
f
(x), expressed in the general form,
i.e., 
y
+ y_{0}
= a_{n}(x
+ x_{0})^{n}
+ a_{n
}_{}_{
}_{1}(x
+ x_{0})^{n}^{
}^{}^{
}^{1}
+
. . . +
a_{1}(x
+ x_{0})
+ a_{0} 
then,
after expanding and reducing the above expression we get
its source function f_{s}(x). 
Inversely,
by plugging the coordinates of translations into the
source polynomial function, i.e.,

y
 y_{0}
= a_{n}(x
 x_{0})^{n}
+ a_{n}_{
}_{}_{
}_{2}(x
 x_{0})^{n}^{
}^{
}^{2}
+
. . . +
a_{2}(x
 x_{0})^{2}
+ a_{1}(x
 x_{0}), 
after
expanding and reducing the above expression, we get given polynomial
f (x). 
Note
that the complete source polynomial has n

1 terms, missing
second and the absolute term. 

Second
method 
The
coefficients of the source polynomial function are related to
corresponding value of the derivative of the given polynomial at x_{0},
like coefficients
of the Taylor polynomial in Taylor's or Maclaurin's formula,
thus 

where,
a_{n}
= a_{n},
a_{n
}_{}_{
}_{1}
= 0,
a_{0}
= f
(x_{0}),
and f
^{(n

k)}(x_{0})
denotes the (n
 k)th
derivative at x_{0}. 
Observe
that
coefficients of the source polynomial function define the value and
the direction of the vertical translation of successive derivatives of given
polynomial that is, 
f
^{(n

k)}(x_{0})
= (n
 k)! a_{n}_{
}_{}_{
}_{k }. 
Further,
the horizontal translation of each successive derivative
corresponds with x_{0}
of the given
polynomial function. 
For
example, the coefficient a_{1}
of the source cubic function
f (x) =
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
since
f ' (x)
=
3a_{3}x^{2}
+ 2a_{2}x
+
a_{1}
and x_{0}
=

a_{2}/(3a_{3})
then 

Thus,
f_{s}(x) =
a_{3}x^{3}
+ a_{1}x
is the source cubic polynomial function, 
where
the coefficient a_{1}
equals the slope of the tangent line at the inflection
point I (x_{0},
y_{0}). 
Further,
as a_{1}
=
f ' (x_{0}),
the coefficient a_{1}
also represents the vertical translation of the first
derivative of the cubic polynomial, that is why the sign
of the product a_{3}a_{1}
is used as the additional condition in the classification
of the cubic polynomial shown later. 




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