The minimum degree of the polynomial f(x) can be determined from its zeros and their multiplicities.
You mentioned that f(x) has zeros at:
1. 2 (multiplicity 3)
2. -5 (multiplicity 3)
3. -5 + 7i (which implies its conjugate -5 - 7i is also a zero)
4. 2 - 4i (which implies its conjugate 2 + 4i is also a zero)
To find the minimum degree of f(x), we need to consider the highest power of x that appears when you expand the polynomial with these zeros and their multiplicities.
For the zeros 2 (multiplicity 3) and -5 (multiplicity 3), we have factors of (x - 2)³ and (x + 5)³, respectively.
For the zeros -5 + 7i and its conjugate -5 - 7i, we have factors of (x - (-5 + 7i))(x - (-5 - 7i)) = (x + 5 - 7i)(x + 5 + 7i), which multiply out to a quadratic term.
For the zeros 2 - 4i and its conjugate 2 + 4i, we have factors of (x - (2 - 4i))(x - (2 + 4i)) = (x - 2 + 4i)(x - 2 - 4i), which also multiply out to a quadratic term.
So, the minimum degree of f(x) will be the sum of the exponents of the highest power terms from these factors:
Minimum degree = 3 + 3 + 2 + 2 = 10
Therefore, the minimum degree of f(x) is 10.