Final answer:
The minimum degree of the polynomial with the given zeros is 8, considering both the multiplicities of the real zeros and the complex zeros along with their conjugate pairs.
Step-by-step explanation:
The question involves finding the minimum degree of a polynomial given its zeros with their specified multiplicities and complex zeros. The zeros given are 4 (multiplicity 3), -2 (multiplicity 1), 5i, and 3+4i.
Since the complex zeros of a polynomial with real coefficients come in conjugate pairs, the minimum degree of the polynomial also includes the conjugates of the complex zeros given, which are -5i and 3-4i.
To find the minimum degree, you add up the multiplicities of the real zeros and count each pair of complex conjugates as two.
The minimum degree is therefore calculated as follows: multiplicity of 4 (3) + multiplicity of -2 (1) + two for 5i and -5i + two for 3+4i and 3-4i. Adding these together: 3 + 1 + 2 + 2 = 8. Thus, the minimum degree of the polynomial is 8.