Euclid conducting a proof (Raphael's fresco School Of Athens, 1511)

The division of polynomials by increasing power order is similar to the usual Euclidean division but in reversed order : the terms of lower degree of the dividend are eliminated first. Some useful algorithms can be deduced from this process, such as series expansion or partial fraction decomposition.

**EUCLIDEAN DIVISION**

It is well known that given two polynomials and on a commutative field K, it is always possible to perform the Euclidean division

The corresponding algorithm consists of dividing by decreasing power order, which means that at each step a division by B is performed on polynomials of decreasing degrees until . Here is an example to recall the method :

Divide by .

Step 1 (degree 3) :

Step 2 (degree 2) :

Step 3 (degree 1) :

**DIVISION BY INCREASING POWER ORDER**

There exists an alternative method to divide polynomials, so called by increasing power order. Here is the theory.

**Theorem**

Let and two polynomials in . We write them and . We assume . Then there exist a unique pair of polynomials in such that

**Proof**

We begin to prove existence by induction. Let be the property for any .

Case . Let , which is allowed since . Then has no constant term, must divide , so there exists such that .

Case under hypothesis that is true. After division of at order , we have with . After division of at order 0, we have with . Now we replace into to get with . Therefore, the property holds with quotient and remainder .

Now we prove unicity. Let and be two solutions. Then we have with and . Clearly, must divide . However, implies that is not an irreducible factor of and thus does not divide . Hence, must divide . Since it is necessary that . It immediately follows .

**SERIES EXPANSION**

The division by increasing power order can help to compute series expansion. Let us consider for example the following series truncated at order 6 :

Step 1 :

Step 2 :

Step 3 :

Hence we have the following expansion

**PARTIAL FRACTION DECOMPOSITION**

The division by increasing power order can also help to compute partial fraction decomposition. Let us consider for example the following fraction :

We divide by as follows.

Step 1 :

Step 2 :

Step 3 :

Step 4 :

Hence :

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