Solve for polynomial Given Zeros

Question

If the two solutions of a polynomial are $4$ and $6 - \sqrt{7}$, how would you solve for the polynomial?

To start would you write the expanded form as $(x-4)$ and $(x-(6-\sqrt{7})$ and then multiply?

Answer 1

Yes. The polynomials for with exactly those two roots are precisely those of the form $c(x-4)^a(x-(6 - \sqrt{7}))^b$, for $a, b \geq 1$ integers and $c$ any real number. With further restrictions (integer polynomials, the degrees of the roots, whatever else), you could narrow that down, potentially to a single solution.