find the polynomial equation with the root x=square root of 3-square root of 5

Answer 1

if √3-√5 is the only root, then clearly
y = x - (√3-√5) will work.

Otherwise, if the roots occur in conjugate pairs, then
y = (x-(√3-√5))(x-(√3+√5)) = x^2 - 2√3 x - 2

Now, since (√3-√5)^2 = 8-2√15, we have
y = (x^2-(8-2√15))(x^2-(8+2√15)) = x^2 - 16x + 4