Sum of roots of polynomial
Algebra
Level
5
Consider the polynomial \(p(x)\) defined as:- \[p(x)=x^4+x^3+2x^2+3x+1\] Let \(r_1,r_2,r_3,r_4\) be the roots of \(p(x)\) then the value of \[ \left( \sum_{i \neq j} \frac{1}{ (1-r_i)(1-r_j) } \right) \left(\sum_{i=1}^4 \frac{1}{1-r_i} \right) \] can be expressed as \(\frac{a}{b}\) where \(gcd(a,b)=1\) find \(a+b\)
Details and assumptions:- I have used the notation \(\sum(\frac{1}{(1-r_i)(1-r_j)}\) to represent the sum of all possible products of the form \(\frac{1}{(1-r_i)(1-r_j)}\) taken two at a time and \(i \neq j\) (i think it also called cyclic sum)
by
Aman Sharma