Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 1 - sqrt(2) and x = 2 + sqrt(7).
one of the factors will be \(x-(1-\sqrt2) = x-1+\sqrt2 \)
think of this as (x-1)+sqrt2
If you introduce another factor as \((x-1)-\sqrt2\)
Then a factor will be
\([(x-1)-\sqrt2][(x-1)+\sqrt2]\\ =(x-1)^2-(\sqrt2)^2\\ =x^2-2x+1-2\\ =x^2-2x-1\)
now do something similar with the \(x=2+\sqrt7 \) root