How do you write the polynomial function with the least degree and zeroes i, 2 - √3?

1 Answer
George C.
Jul 17, 2015

f(z) = (z-i)(z-2+sqrt(3))f(z)=(z−i)(z−2+√3)

= z^2-(i+2-sqrt(3))z+i(2-sqrt(3))=z2−(i+2−√3)z+i(2−√3)

If you want rational coefficients then:

g(z) = (z-i)(z+i)(z-2+sqrt(3))(z-2-sqrt(3))g(z)=(z−i)(z+i)(z−2+√3)(z−2−√3)

=(z^2+1)(z^2-4z+1)=z^4-4z^3+2z^2-4z+1=(z2+1)(z2−4z+1)=z4−4z3+2z2−4z+1

Explanation:

We are dealing with conjugates here.

To get a real number from ii, multiply it by +-i±i.

To get a rational number from 2-sqrt(3)2−√3, multiply it by 2+sqrt(3)2+√3.

Basically, from the perspective of the rational numbers QQ, the numbers i and -i are indistinguishable and the numbers sqrt(3) and -sqrt(3) (and thus 2+sqrt(3) and 2-sqrt(3)) are only distinguishable by ordering - not by 'algebraic' properties. That is, if one of these non-rational numbers is a root of a polynomial with rational coefficients, then its conjugate must also be.

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