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Let *a* and *b* be two univariate polynomials in
[*x*].
We assume that *b* is **monic** and has degree *n* > 0
and that *a* has degree *n* + *k* whith *k* 0.
We define
*a* = *a*_{n+k}*x*^{n+k} + ^{ ... } + *a*_{n}*x*^{n} + ^{ ... } + *a*_{1}*x* + *a*_{0} and
*b* = *b*_{n}*x*^{n} + ^{ ... } + *b*_{1}*x* + *b*_{0}.
We are interesting in computing the remainder *r*
of the Euclidean division of *a* by *b* (regarding them
as polynomials in
[*x*]), by a modular method.
Let *p* be a prime number and be the application that
maps every integer *z* with its residue class
(*z*) in
/*p*.
We extend this application to a map denoted again,
from
[*x*] to
/*p*[*x*], such that *x* is mapped to *x*.
- Explain why the remainder
*r* belongs to
[*x*], that is
no fractions appears during the computations.
- Consider
*a* = *x*^{2} + 7*x* + 1 and *b* = *x* + 4. Compute *q* and *r*.
- Consider
*p* = 3. Compute
(*a*) and
(*b*).
Then compute the quotient and the remainder of
(*a*) by
(*b*).
- In the general case, what is the relation between
*r* and the remainder
of
(*a*) by
(*b*)?
- Design a modular method to compute
*r*.

**Answer 4**
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*

**Answer 5**
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*Marc Moreno Maza *

2006-01-09