Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial $f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where $i \in {1,2,3,...,8}$. Find the possible values of $a_1$.
Source: NMTC 2023 Finals P3 (Junior)
Tags: algebra, polynomial, nmtc
Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial $f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where $i \in {1,2,3,...,8}$. Find the possible values of $a_1$.