YWANG Generalization
The YWANG Generalization for Polynomial Factorization (discovered by AoPS user Yufanwang) is a generalization of polynomial factorization. It differs from traditional techniques (such as synthetic division) in that the coefficients of each term is explicitly defined, and is applicable to the factoring of more than one element of the form 
at the same time.
Theorem
As per traditional techniques, we limit our constants to be within 
(i.e. we will define 
, 
, and 
). Then, for a given 
and its factors 
we have ![\[f(x)=\prod^i_{k=0}(x-a_k)\sum^{n-1}_{m=0}\gamma_{i, m}x^{n-m-1}\]](http://latex.artofproblemsolving.com/e/8/f/e8f9c23e4e88aee09fd6961cde4114ca3da017c9.png)
where 
designates the 
th term of the 
th YWANG gamma sequence. Each term of the gamma sequence is defined as follows: ![\[\gamma_{i,j}=\sum_{k=0}^j\left(c_k\sum_{\substack{b_1\cdots b_i\in [0,1]\cap\mathbb{Z} \\ \sum_{l=1}^i b_l = k}} \left(\prod_{l=1}^i a_l^{b_l}\right)\right).\]](http://latex.artofproblemsolving.com/6/c/9/6c92a11271c57a2d625dfa52397fae876fee616a.png)
This result can be derived from strong division with synthetic division or by taking the Taylor series of ![\[\frac{\sum_{m=0}^n c_m x^{n-m}}{\prod_{k=0}^i (x-a_k)}\]](http://latex.artofproblemsolving.com/9/0/2/9025ff5b3876b2e9d134789e584fccb9aa66a16a.png)
at 
. A rigorous proof with both methods will be added in due time.
YWANG Gamma Sequences
The th YWANG gamma sequence is given in the form We will consider the th YWANG gamma sequence solved if there are not nested sums/products within the final expression.
Solved YWANG Gamma Sequences
A list of currently solved YWANG gamma sequences follows. 
This list will be updated as further gamma sequences are solved.
