Final answer:
The number of complex zeros in a polynomial function is given by the Fundamental Theorem of Algebra, which states that a polynomial of degree n will have exactly n complex roots. To find how many complex roots a given polynomial has, one can factor the polynomial or use various numerical methods, keeping in mind complex roots appear in conjugate pairs if the polynomial has real coefficients.
Step-by-step explanation:
To find the number of complex zeros in the polynomial function f(x) = x^4 + 4x^3 - 4x^2 + 44x - 195, we need to understand the Fundamental Theorem of Algebra and the Conjugate Pairs Theorem. According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n complex roots which include both real and non-real (complex) zeros.
To determine the exact number and nature of these zeros, one would typically factor the polynomial or use numerical methods. If a polynomial with real coefficients has complex zeros, they will occur in conjugate pairs due to the Conjugate Pairs Theorem. This means if a + bi is a zero, then its conjugate a - bi will also be a zero of the polynomial.
In this case, since the polynomial given is a quartic function, which is a fourth degree polynomial, it will have up to 4 zeros. If the problem implies we have one real zero given which is -3, then we can find the remaining zeros by dividing the polynomial by (x + 3) to reduce the degree of the polynomial and then solve the resulting cubic equation to find the remaining zeros.