Question

Given: Polynomial P(x) of degree 6

Given: x=3 is a zero for the Polynomial above

List all combinations of real and complex zeros, but do not consider multiplicity for the zeros.

Answer 1

P(x) is a polynomial of degree 6 and x = 3 is a zero of P(x).

The following combinations are possible for the remaining 5 zeroes of P(x):

1. All the remaining 5 zeroes of P(x) are real;

2. As complex zeros occur in conjugate pairs, 2 of the remaining 5 zeroes of P(x) are complex and the remaining 3 zeroes of P(x) are real;

3. As complex zeros occur in conjugate pairs, 4 of the remaining 5 zeroes of P(x) are complex and the remaining 1 zero of P(x) is real.

Thus, the only combinations possible are:

i. All 6 zeroes of P(x) are real;

ii. 2 zeroes of P(x) are complex and 4 zeroes of P(x) are real;

iii. 4 zeroes of P(x) are complex and 2 zeroes of P(x) are real;