How do you write a polynomial in standard form given the zeros x=−3i and √3?
1 Answer
According to what you want, the simplest polynomials are:
x2+(3i−√3)x−3√3i
x3−√3x2+9x−9√3
x4+6x2−27
Explanation:
A polynomial of lowest degree with these zeros is:
(x+3i)(x−√3)=x2+(3i−√3)x−3√3i
Typically we would be interested in the polynomial having Real coefficients. If so, then any non-Real Complex zeros occur in Complex conjugate pairs. Hence
(x−3i)(x+3i)(x−√3)
=(x2+9)(x−√3)
=x3−√3x2+9x−9√3
If we also want the polynomial to have rational coefficients, then any irrational zeros of the form
Hence the simplest polynomial with the given zeros and rational coefficients is the quartic:
(x−3i)(x+3i)(x−√3)(x+√3)
=(x2+9)(x2−3)
=x4+6x2−27
