Question

Is it possible for a polynomial of degree 5 to have only 2 of its zeros be real? Why or why not?

a) Yes, because a degree 5 polynomial can have any combination of real and complex zeros.
b) No, because a polynomial of degree 5 must have all its zeros be real.
c) Yes, but only if the polynomial has complex conjugate pairs of zeros.
d) No, because the number of real zeros of a polynomial is always odd for odd-degree polynomials.

Answer 1

A polynomial of degree 5 can have 2 real zeros if the remaining zeros are complex conjugate pairs.

Step-by-step explanation:

The correct option is c) Yes, but only if the polynomial has complex conjugate pairs of zeros.

A polynomial of degree 5 can have any combination of real and complex zeros. However, if the polynomial has 2 real zeros, then the other 3 zeros must be complex conjugate pairs.

For example, consider the polynomial f(x) = (x - 1)(x - 2)(x^2 + 1)(x^2 + 2x + 5). This polynomial has 2 real zeros (x = 1 and x = 2) and 3 complex zeros (x = i, x = -i, and x = -1 + 2i).