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The '''YWANG Generalization for Polynomial Factorization''' (discovered by AoPS user [https://artofproblemsolving.com/community/user/593414 Yufanwang]) is a generalization of polynomial factorization. It differs from traditional techniques (such as [[Synthetic division|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 <math>(x-a_i)</math> at the same time.

==Theorem==
As per traditional techniques, we limit our constants to be within <math>\mathbb{R}</math> (i.e. we will define <math>c_0, c_1, \cdots, c_n \in \mathbb{R}</math>, <math>i\in \mathbb{N}</math>, and <math>n\in \mathbb{Z}</math>). Then, for a given <math>f(x)=c_0x^n+c_1x^{n-1}+\cdots+c_{n-1}x+c_n=\textstyle\sum^n_{m=0}c_mx^{n-m}</math> and its factors <math>(x-a_1)(x-a_2)\cdots(x-a_i)</math> we have <cmath>f(x)=\prod^i_{k=0}(x-a_k)\sum^{n-1}_{m=0}\gamma_{i, m}x^{n-m-1}</cmath> where <math>\gamma_{i,m}</math> designates the <math>m</math>th term of the <math>i</math>th [[YWANG Generalization#YWANG Gamma Sequences|YWANG gamma sequence]]. Each term of the gamma sequence is defined as follows: <cmath>\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).</cmath> This result can be derived from strong division with [[Synthetic division|synthetic division]] or by taking the [[Taylor polynomial|Taylor series]] of <cmath>\frac{\sum_{m=0}^n c_m x^{n-m}}{\prod_{k=0}^i (x-a_k)}</cmath> at <math>x=\infty</math>. A rigorous proof with both methods will be added in due time.

==YWANG Gamma Sequences==
The <math>i</math>th '''YWANG gamma sequence''' is given in the form <cmath>\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).</cmath> We will consider the <math>i</math>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. <cmath>\begin{align*}
\gamma_{1,j} &= \sum_{k=0}^j c_k a_1^{j-k} \\
\gamma_{2,j} &= \sum_{k=0}^j c_k \frac{a_1^{j-k}-a_2^{j-k}}{a_1-a_2} \\
\gamma_{3,j} &= \sum_{k=0}^j c_k \frac{a_1^{2(j-k)-1}(a_3-a_2)+a_2^{2(j-k)-1}(a_1-a_3)+a_3^{2(j-k)-1}(a_2-a_1)}{(a_1-a_2)(a_2-a_3)(a_3-a_1)}
\end{align*}</cmath> This list will be updated as further gamma sequences are solved.