Polynomial function that has the zeros -5 and 4i

The polynomial function is:

Explanation:

In this exercise, we will use two important theorems:

Fundamental Theorem of Algebra: We can factor completely any polynomial with real number coefficients over the field of complex numbers.Complex Conjugate Root Theorem: If a polynomial in one variable has real coefficients, and is a root of that polynomial being real numbers, then its complex conjugate is also a root of the polynomial function.

In this case we have the following roots:

According to the Complex Conjugate Root Theorem:

Hence, we can write the function as: