

ALGEBRA


Polynomial
functions' properties

Every
polynomial has its initial position at the origin of the
coordinate system. A polynomial function written in
general form represents translation of its source
(original) function

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,

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}.

1)
By plugging 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_{2}(x
+ x_{0})^{2}
+
a_{1}(x
+ x_{0})
+ a_{0}

and after expanding and reducing the above expression we get
its source function f_{s}(x).

2)
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}),

and after
expanding and reducing the above expression, we get given polynomial
f (x).

Moreover, the
coefficients a
of the source polynomial 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,


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}.

:: Sigma
notation of the polynomial function 
Sigma
notation of the polynomial, where the coefficients a
of its source function are given by a recursive formula, 

while,
for k
= 0, a_{n}
= a_{n},
for k
= 1, a_{n1}
= 0
and for k
= n, a_{0}
= f
(x_{0})
= y_{0}. 


Graphs of polynomial functions

Polynomial
functions are named in accordance to their degree.

::
Zero polynomial
f
(x)
= 0 
The
constant polynomial f
(x)
= 0
is called the zero polynomial and
is graphically represented by the xaxis. 
::
Constant function
f (x)
= a_{0} 
A
polynomial of degree 0, f
(x)
= a_{0},
is called a constant function,
its
graph is a horizontal line with the yintercept
a_{0}. 



::
Linear
function, the first degree polynomial f
(x)
= a_{1}x + a_{0}

y
= a_{1}
x + a_{0}
or y
= m
x + c 
or y
= a_{1}(x

x_{0})
or y

y_{0}
= a_{1}x,

where
x_{0}
= _{
}
a_{0 }/a_{1}
and/or y_{0}
= a_{0}
are the xintercept
and the yintercept
respectively, and where the slope of the
line a_{1}
= _{
}m,


By
setting x_{0}
= 0 or y_{0}
= 0
obtained is 
the
source linear function,
y
= a_{1}x. 



::
The roots of a
polynomial or zero function
values, xintercepts

A
zero of a function is a value of the argument (of a function) at
which the value of the function is zero. 
Therefore,
zeros are values of the argument x
that satisfy or solve the equation f
(x)
= 0
or y
= 0. 
The
zeros (roots) of a function correspond to the xintercepts
of the graph. 
An
xintercept
is the point (x,
0)
where the graph of the function touches or crosses the xaxis. 

::
Linear
equation

Thus, the
solution of the equation f
(x)
= 0
or y
= 0,
that
is 
a_{1}
x + a_{0}
= 0
or m
x + c
= 0, 
is the
zero of the linear function, the xintercept or
the root of the first degree polynomial 
















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