I need some assistance with this

Question

Suppose that R (x) is a polynomial of degree 7 whose coefficients are real numbers. Also, suppose that R (x) has the following zeros: -1, -3i, 2i

Answer the following:

(a) Find another zero of R (x).

(b) What is the maximum number of real zeros that R (x) can have?

(c) What is the maximum number of nonreal zeros that R (x) can have?

Answer 1

a) Since the polynomial has real coefficients we know that complex roots come in pairs. Since -3i is a root, its conjugate, 3i will also be a root.

b) Since the polynomial has degree 7 and since we know that we have at least these complex roots -3i, 3i, 2i, -2i the maximum number of real roots is 7-4=3.

c) Again, since the polynomial has degree 7 and we already know 5 of its roots, -1, -3i, 3i, -2i, 2i the maximum number of non real roots is 7-1=6.