

Polynomial and/or Polynomial
Functions and Equations 
Sigma
notation of the polynomial 
Coefficients of the source
polynomial in the form of a recursive formula 
Coefficients
of the source polynomial function are related to its derivative
at x_{0} 





Sigma
notation of the polynomial 
Coefficients of the source
polynomial shown by a recursive formula 
Therefore,
the polynomial f
(x) =
y = a_{n}x^{n}
+ a_{n}_{1}x^{n}^{}^{1}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
¼
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
we can
write as



while, for k = 0,
a_{n}
=
a_{n}, 

and from a_{n}_{}_{ k}, for
k =
n,
a_{0}
=
f
(x_{0})
= y_{0}. 

The
expanded form of the above
sum is 
y
 y_{0}
= a_{n}(x
 x_{0})^{n}
+ a_{n}_{}_{2}(x
 x_{0})^{n}^{}^{2}
+ ¼
+
a_{2}(x
 x_{0})^{2}
+ a_{1}(x
 x_{0}) 
where
x_{0}
and y_{0}
are coordinates of translations
of the graph of the source polynomial 
f_{s}(x)
= a_{n}x^{n}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+ ¼
+
a_{2}x^{2}
+ a_{1}x

in
the direction of the xaxis
and the yaxis
of a Cartesian coordinate system. 
Let's
finally mention the
main property of a polynomial that shows the nature of the
revealed theory the best. 
An
nth
degree polynomial function and all its successive derivatives to the (n

1)th
order, have constant
horizontal translation x_{0}. 

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


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

k)}(x_{0})
denotes the (n
 k)th
derivative at x_{0}. 
Such
for example, the coefficient a_{1}
of the source cubic of
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 









Functions
contents C 



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