Solve for polynomial Given Zeros

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?

• Yes, that gives you a polynomial with those roots. – Alex J Best Nov 14 '18 at 23:47

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.
• For what it's worth: You can't have $6-\sqrt7$ be a root of a polynomial with integer (or rational) coefficients without also having $6+\sqrt7$ as a root. – Arthur Nov 14 '18 at 23:50