Find a polynomial with integer coefficients such that (sqrt3 + sqrt5) is a root of the polynomial

Answer 1

Going back over some previous posts, I noticed you had posted this same question earlier, but for that you had
3 + √5 as a root.

I will answer it as if that was the right question.

If 3+√5 is a root, then its conjugate 3-√5 must also be a root, or else a radical will show up in the expansion.

so the polynomial is
(x - (3+√5))(x- (3-√5))
= (x - 3 - √5)(x - 3 + √5)
= x^2 - 3x + x√5 - 3x + 9 - 3√5 - x√5 + 3√5 - 5
= x^2 - 6x + 4