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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 x0 |
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Sigma
notation of the polynomial |
Coefficients of the source
polynomial shown by a recursive formula |
Therefore,
the polynomial f
(x) =
y = anxn
+ an-1xn-1
+ an-2xn-2
+
¼
+
a2x2
+
a1x + a0 |
we can
write as
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while, for k = 0,
an
=
an, |
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and from an- k, for
k =
n,
a0
=
f
(x0)
= y0. |
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The
expanded form of the above
sum is |
y
- y0
= an(x
- x0)n
+ an-2(x
- x0)n-2
+ ¼
+
a2(x
- x0)2
+ a1(x
- x0) |
where
x0
and y0
are coordinates of translations
of the graph of the source polynomial |
fs(x)
= anxn
+ an-2xn-2
+ ¼
+
a2x2
+ a1x
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in
the direction of the x-axis
and the y-axis
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
n-th
degree polynomial function and all its successive derivatives to the (n
-
1)-th
order, have constant
horizontal translation x0. |
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Coefficients
of the source polynomial function are related to its derivative
at x0 |
The
coefficients of the source polynomial are related to
corresponding value of its derivative at x0
like the coefficients
of the Taylor polynomial in Taylor's or Maclaurin's formula,
thus |
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since,
an
= an,
an-1
= 0,
a0
= f
(x0),
and where f
(n
-
k)(x0)
denotes the (n
- k)th
derivative at x0. |
Such
for example, the coefficient a1
of the source cubic of
f(x) =
a3x3
+
a2x2
+
a1x + a0 |
since
f ' (x)
=
3a3x2
+ 2a2x
+
a1
and x0
=
-
a2/(3a3)
then |
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Functions
contents C |
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© 2004 - 2020, Nabla Ltd. All rights reserved. |