Polynomial function

Question

Find a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros. Square root of three, 4i

Answer 1

First, let's start with the fact that the zeros of a polynomial divide it (in the complex numbers). That is, for a polynomial with zeros sqrt(3) and 4i, the polynomials x-sqrt(3) and x-4i divide it. But, we gotta have rational coefficients, so we need to get rid of the square root and the i terms.

To do that, the identity (a+b)(a-b)=a^2-b^2 is pretty useful, since (sqrt(3))^2=3 and (-4i)^2=-16. So the lowest degree polynomial with rational coefficients is a polynomial of degree 4, namely:

(x-sqrt(3))(x+sqrt(3))(x-4i)(x+4i)=(x^2-3)(x^2+16)=x^4+13x^2-48.

I'll leave it to you to do the checking to see that sqrt(3) and 4i are zeros of this polynomial.