Suppose that [tex]\( R(x) \)[/tex] is a polynomial of degree 8 whose coefficients are real numbers. Also, suppose that [tex]\( R(x) \)[/tex] has the following zeros: [tex]\( 3, -5, 9, -2-4i \)[/tex].

Answer the following:

(a) Find another zero of [tex]\( R(x) \)[/tex].
[tex]\[ -2+4i \][/tex]

(b) What is the maximum number of real zeros that [tex]\( R(x) \)[/tex] can have?
[tex]\[ \square \][/tex]

(c) What is the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have?
[tex]\[ \square \][/tex]



Answer :

Let's solve each part of the question step-by-step.

### Given:
- [tex]\( R(x) \)[/tex] is a polynomial of degree 8 with real coefficients.
- [tex]\( R(x) \)[/tex] has the zeros: [tex]\( 3 \)[/tex], [tex]\( -5 \)[/tex], [tex]\( 9 \)[/tex], and [tex]\( -2 - 4i \)[/tex].

### (a) Find another zero of [tex]\( R(x) \)[/tex]:

Since [tex]\( R(x) \)[/tex] has real coefficients, any non-real (complex) zeros must come in conjugate pairs. This ensures that the polynomial remains with real coefficients. Given that [tex]\( -2 - 4i \)[/tex] is a zero, its conjugate [tex]\( -2 + 4i \)[/tex] must also be a zero.

Thus, another zero of [tex]\( R(x) \)[/tex] is:
[tex]\[ -2 + 4i \][/tex]

### (b) What is the maximum number of real zeros that [tex]\( R(x) \)[/tex] can have?

- We know that the polynomial [tex]\( R(x) \)[/tex] is of degree 8, meaning it can have at most 8 zeros in total.
- Complex roots occur in conjugate pairs.

Given the zero [tex]\( -2 - 4i \)[/tex] and its conjugate [tex]\( -2 + 4i \)[/tex], we already have one pair of complex zeros, which accounts for 2 of the 8 possible zeros.

The remaining zeros must be real since the degree of the polynomial is 8.

Thus, the maximum number of real zeros is:
[tex]\[ 8 (total degree) - 2 (complex zeros) = 6 \][/tex]

### (c) What is the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have?

- As mentioned, [tex]\( R(x) \)[/tex] is of degree 8.
- We already identified that there can be at most 6 real zeros.

The total number of zeros must be 8, so the remaining can be nonreal.

Thus, the maximum number of nonreal zeros is:
[tex]\[ 8 (total degree) - 6 (max number of real zeros) = 2 \][/tex]

### Summary:

(a) Another zero of [tex]\( R(x) \)[/tex] is:
[tex]\[ -2 + 4i \][/tex]

(b) The maximum number of real zeros that [tex]\( R(x) \)[/tex] can have is:
[tex]\[ 6 \][/tex]

(c) The maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have is:
[tex]\[ 2 \][/tex]