Task (Working with Polynomials, 5 Points)
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Consider a polynomial
, given by
with
and
. Show that
can also be expressed as a polynomial in
and
by specifying the coefficients in

.
For
,is
. Solving the system
for
, we get
and
. We substitute these into
. Using the Binomial theorem, we get
We then perform an index transformation, replacing the summation over
with a summation over
. Since
,
ranges from
to
. For a fixed
, we consider
values between
and
, where
, so
. Hence,
ranges from
to
. We get
We then switch the order of summation. We have
and for a fixed
,
, which means
,
resulting in
Finally, we perform another index transformation. We replace
with
.
ranges from
to
. For a fixed
,
we have
. Therefore, we consider only values of
for which
, i.e.,
.
We get
Thus,
Translation and Version Control
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