(a) Complex zeros of a polynomial come in pairs.
If a + bi is a zero of a polynomial then its conjugate a - bi is also a zero of the polynomial.
The given complex zeros of R(x) are 1 + 3i and -2i.
1 - 3i is the conjugate of 1 + 3i.
Hence, another zero of R(x) is 1 - 3i
b)
Since the polynomial R(x) is of order 13 then R(x) must have 13 zeros.
The given complex zeros of R(x) are 1 + 3i and -2i. We also know that the conjugates of 1 + 3i and -2i are zeros of R(x). Hence, R(x) has at least 4 complex roots
Hence, the maximum number of real zeros of R(x) is (13 -4).
The maximum number of real zeros of R(x) is 9
c) Let the maximum number of nonreal zeros (complex roots) be n
Complex roots come in pairs. Therefore, n must be even.
Hence, n ≤ 13 - 1 = 12
n ≤ 10
We have been given a real zero of R(x), 3 ( With the multiplicity of 4).
12 - 4 = 8
Therefore,
n ≤ 8.
Hence the maximum number of nonreal zeros of R(x) is 8
