

Polynomial and/or Polynomial
Functions and Equations 
Polynomial functions 
The source or the original polynomial function 
Translating
(parallel shifting) of the source polynomial function 
Coordinates of translations and their role in the polynomial
expression 
Roots or zeros of
polynomial function 





Polynomial functions 
The
source
or the original polynomial function 
A
polynomial f
(x)
= a_{n}x^{n}
+ a_{n}_{}_{1}x^{n}^{}^{1}
+
.
. . +
a_{1}x
+ a_{0}
of degree n >
1, consisting
of n
+ 1 terms, shown 
graphically, represents translation of its
source (original) function in the direction of the coordinate
axes. 
The complete source polynomial function 

f_{s}(x)
= a_{n}x^{n}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+ a_{1}x



has
n
 1 terms
lacking second and the constant term, since its coefficients, a_{n}_{}_{1
}=
0
and a_{0
}=
0
while
the leading coefficient a_{n},
remains unchanged. 
Therefore,
the source polynomial function passes through the
origin. 
A
coefficient a_{i
}of
the source function is expressed by the coefficients of the general
form. 
A
coefficient a_{i
}of
the source polynomial can be calculated like
coefficients of the Taylor polynomial 

where
f ^{(}^{i}^{
)}^{
}(x_{0})
denotes ith derivative at
x_{0}. 

Translating
(parallel shifting) of the source polynomial function 
Thus,
to obtain the graph of a given polynomial function f
(x)
we translate (parallel shift)
the
graph of its source function in the direction of the x^{}axis
by x_{0}
and in the direction of the yaxis
by y_{0}. 
Inversely,
to put a given graph of the polynomial function beck to the
origin, we translate it in the opposite direction, by taking the
values of the
coordinates of translations with opposite sign. 

Coordinates of translations
and their role in the polynomial expression 
The
coordinates of translations we calculate using the formulas, 

Therefore,
by plugging the coordinates of translations into
the source polynomial function f_{s}(x),
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}) 


and
by expanding above expression we get the polynomial function in
the general form. 

Inversely, by plugging the coordinates of translations into a given polynomial function
f (x), that is 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} 


and
after expanding and reducing above expression we get its source polynomial function. 
Note
that in the above expression the signs of
the coordinates of translations are already changed. 
Therefore,
each polynomial missing second term (a_{n}_{1}
=
0),
represents a source polynomial whose graph is translated in
the direction of the yaxis
by y_{0}
= a_{0}. 

Roots or zeros of
polynomial function 
The
zeros of a polynomial function are the values of x
for which the function equals zero. 
That
is, the solutions of the equation f
(x)
= 0,
that are called roots of the polynomial, are the zeros of the
polynomial function or the xintercepts of its
graph in a coordinate plane. 
At
these points the graph of the polynomial function cuts or
touches the xaxis. 
If
the graph of a polynomial intersects with the xaxis
at (r,
0), or x
= r
is a root or zero of a polynomial, then 
(x

r)
is a factor of that polynomial.

Every
polynomial of degree n
has exactly n
real and/or complex zeros. 
An
nth
degree polynomial has at most n
real zeros. 
Some
of the roots may be repeated. The number of times a root is
repeated is called multiplicity or
order of the root. 
The
number r_{i}
is a root of the
polynomial f
(x)
if and only if
f (x)
is divisible by
(x

r_{i}).

Therefore,
a polynomial and/or polynomial function with
real coefficients can be expressed as a product of its leading
coefficient a_{n
}and
n
linear factors
of the form (x
 r_{i}),
where r_{i}_{
}denotes its real root and/or complex root, 
f
(x)
= a_{n}x^{n}
+ a_{n}_{}_{1}x^{n}^{}^{1}
+
.
. . +
a_{1}x
+ a_{0}
= a_{n}(x
 r_{1})(x
 r_{2})
.
. . (x
 r_{n}).

Thus, finding the roots of a
polynomial f
(x)
is equivalent to finding its
linear divisors or is equivalent to polynomial
factorization into linear factors. 








Precalculus contents
E 



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