The polynomial equation \[x^3 + bx + c = 0,\]where $b$ and $c$ are rational numbers, has $3-\sqrt{7}$ as a root. It also has an integer root. What is it?

Why I'm confused, so in my last repost, melody, a great answered wanted me to find the root with the conjugate, with the factor theorem,

its true you are allowed to do that if the coefficients are real. I obviously did that and I got $x^2-6x+2$. Do I use synthetic division and divide it? I'm really confused since I know synthetic division but I don't know with variables...

Guest Jan 12, 2021