Answer:
(a) -4 - 5i
(b) 6
(c) 4
Explanation:
Part (a)
According to the Complex Conjugate Root Theorem, complex zeros of a polynomial with real coefficients always come in conjugate pairs. So, if a complex number is a zero of a polynomial, its complex conjugate is also a zero.
The complex conjugate of a complex number is obtained by changing the sign of its imaginary part while keeping the real part unchanged. Therefore, as -4 + 5i is a zero of R(x), then -4 - 5i is also a zero.
Part (b)
The maximum number of real zeros for a polynomial is equal to its degree. Since R(x) is of degree 8, and we have already determined that it has a minimum of two nonreal zeros, it can have a maximum of 6 real zeros.
Part (c)
As complex zeros of a polynomial with real coefficients always come in conjugate pairs. Given that there are at least 3 real zeros (-3, 8 and -9), and R(x) is of degree 8, it can have a maximum of 4 nonreal zeros.