posted by nancy .
find a polynomial with integer coefficients such that square root of 3 + square root of 5 is a root of the polynomial
There are an infinite number of polynomials with that as one solution, but one of the simplest would be the one for which 3 - sqrt5 is also a solution. That would be
(x -3 -sqrt5)(x -3 +sqrt5)= 0
(x-3)^2 -5 = 0
x^2 -6x +4 = 0