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.
−4 −4i, 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 non-real zeros that R(x) can have?

Answer 1

The conjugates of the given complex zeros will provide another zero for R(x). The maximum number of real zeros that R(x) can have is equal to its degree. The maximum number of non-real zeros can be calculated using the degree and the number of real zeros.

Step-by-step explanation:

(a) To find another zero of R(x), we need to consider the conjugates of the given complex zeros. So, the conjugates of -4-4i and 2i are -4+4i and -2i, respectively. Since R(x) has real coefficients, this means that the complex zeros occur in conjugate pairs. Therefore, another zero of R(x) is -4+4i.

(b) The maximum number of real zeros that R(x) can have is equal to its degree, which is 7.

(c) The maximum number of non-real zeros that R(x) can have is also equal to its degree minus the number of real zeros. So, in this case, the maximum number of non-real zeros would be 7 - 4 = 3.