How do I write the polynomial function of 5/3 and 2i?

Question

My worksheet is asking me to write a polynomial function of least degree with integral coefficients that has the given zeros. I don’t remember how to write polynomial functions with imaginary numbers.

Answer 1

The complex conjugate of a complex zero must also be a zero, so you will have 3 zeros:

5/3

2i

-2i

f(x) = A(x - 5/3)(x - 2i)(x + 2i) = A(x - 5/3)(x² + 4)

To ensure integer coefficients, A must be real and must be a multiple of 3. For example, A can be 3 and that meets the conditions of the problem.

f(x) = 3(x - 5/3)(x² + 4)

More generally,

f(x) = 3n(x - 5/3)(x² + 4),

where n is any integer.