Suppose that [tex]$R(x)$[/tex] is a polynomial of degree 7 whose coefficients are real numbers. Also, suppose that [tex]$R(x)$[/tex] has the following zeros:
[tex]$
-i, \quad -1-5i, \quad 3-2i
$[/tex]

Answer the following:

(a) Find another zero of [tex]$R(x)$[/tex].
[tex]$ \square $[/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 :

Given the polynomial [tex]\( R(x) \)[/tex] of degree 7 with real coefficients and the zeros provided:

[tex]\[ -i, \quad -1-5i, \quad 3-2i \][/tex]

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

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

Since the coefficients of [tex]\( R(x) \)[/tex] are real, the non-real zeros must occur in conjugate pairs. This means that for each non-real zero [tex]\( a + bi \)[/tex], [tex]\( a - bi \)[/tex] must also be a zero.

The given zeros are:
- [tex]\( -i \)[/tex]
- [tex]\( -1 - 5i \)[/tex]
- [tex]\( 3 - 2i \)[/tex]

Their conjugates are:
- [tex]\( i \)[/tex] (conjugate of [tex]\( -i \)[/tex])
- [tex]\( -1 + 5i \)[/tex] (conjugate of [tex]\( -1 - 5i \)[/tex])
- [tex]\( 3 + 2i \)[/tex] (conjugate of [tex]\( 3 - 2i \)[/tex])

Therefore, the additional zeros must be:
[tex]\[ i, \quad -1 + 5i, \quad 3 + 2i \][/tex]

So, the additional zeros are:
[tex]\[ i, \quad -1 + 5i, \quad 3 + 2i \][/tex]

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

The polynomial [tex]\( R(x) \)[/tex] is of degree 7, meaning it can have a total of 7 zeros (real or complex). We already identified 6 non-real zeros (3 given and their 3 conjugates).

The number of non-real zeros:
[tex]\[ 6 \][/tex]

Since the polynomial is of degree 7, the remaining [tex]\( 7 - 6 = 1 \)[/tex] zero must be real.

Maximum number of real zeros:
[tex]\[ 1 \][/tex]

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

From part (b), we have already determined that out of the 7 zeros, 6 are non-real because they appear in conjugate pairs. This is the maximum number of non-real zeros for this polynomial.

Maximum number of nonreal zeros:
[tex]\[ 6 \][/tex]

To summarize:
- (a) Find another zero of [tex]\( R(x) \)[/tex].
[tex]\[ i, -1 + 5i, 3 + 2i \][/tex]
- (b) What is the maximum number of real zeros that [tex]\( R(x) \)[/tex] can have?
[tex]\[ 1 \][/tex]
- (c) What is the maximum number of nonreal zeros that [tex]\( R(x) \)[/tex] can have?
[tex]\[ 6 \][/tex]