Linear factors theorem and conjugate zeros theorem

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+i, -1+3i, -4i

 

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

The polynomial is degree 7 so there are 7 roots.

 

For any nonreal root, the conjugate is also a root so we have roots at:

 

4+i, 4-i, -1+3i, -1-3i, -4i, 4i

 

This accounts for 6 of the 7 roots.

Since there are 7 roots, the maximum number of real zeroes is 1

 

The maximum number of nonreal zeroes is 6 because another nonreal zero

would have to include it's conjugate, thus taking the number of roots to 8.