Question

How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are -2, -2, 3, -4i?

Answer 1

Construct it from linear terms including the given roots

Explanation:

`x` minus each of the zeroes is equal to zero, so `f(x)=(x+2)^2(x-3)(x+4i)` does the job and has no more degrees than needed - assuming that the question-setter requires the repeated root at `x=-2`. If they don't, then `f(x)=(x+2)(x-3)(x+4i)` is better.

This multiplies out to be
`f(x)=x^3-(1-4i)x^2-(6+4i)x-24i`
in the non-repeated root case and
`f(x)=x^4+(1+ri)x^3-(8-4i)x^2-(12+32i)x-48i`
if we include the root at `x=-2` twice.