Here is a non-trivial example of factorization using symmetric polynomials.

Question: Factorize into five factors the following expression:

.

Solution: Observe that the given expression is symmetric, homogeneous in degree 6.

Let

If we put one by one in the above expression E, the expression E collapses to zero. Hence, by the factor theorem, the following third degree, homogeneous, symmetric expression is a factor of E:

.

The above is just third degree so let us assume that:

.

Step 1. Put so that we get

Step 2. Put so that we get

Hence, solving Step 1 and Step 2, we get .

Step 3. Put so that we get .

Hence, is factorized into five factors as follows:

which can be further factorized as:

.

More later,…

Please send your suggestions, comments, etc.

Nalin Pithwa