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.
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:
Please send your suggestions, comments, etc.