2.9k views
0 votes
Suppose that R(x) is a polynomial of degree 13 whose coefficients are real numbers. also, suppose that R(x) has the following zeros. answer the following.edit: if possible please double check the answers just to be safe.

Suppose that R(x) is a polynomial of degree 13 whose coefficients are real numbers-example-1

1 Answer

4 votes

(a) Complex zeros of a polynomial come in pairs.

If a + bi is a zero of a polynomial then its conjugate a - bi is also a zero of the polynomial.

The given complex zeros of R(x) are 1 + 3i and -2i.

1 - 3i is the conjugate of 1 + 3i.

Hence, another zero of R(x) is 1 - 3i

b)

Since the polynomial R(x) is of order 13 then R(x) must have 13 zeros.

The given complex zeros of R(x) are 1 + 3i and -2i. We also know that the conjugates of 1 + 3i and -2i are zeros of R(x). Hence, R(x) has at least 4 complex roots

Hence, the maximum number of real zeros of R(x) is (13 -4).

The maximum number of real zeros of R(x) is 9

c) Let the maximum number of nonreal zeros (complex roots) be n

Complex roots come in pairs. Therefore, n must be even.

Hence, n ≤ 13 - 1 = 12

n ≤ 10

We have been given a real zero of R(x), 3 ( With the multiplicity of 4).

12 - 4 = 8

Therefore,

n ≤ 8.

Hence the maximum number of nonreal zeros of R(x) is 8

User Davecave
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories