Given the polynomial function with rational coefficients and the zeros [tex]\(i\)[/tex] and [tex]\(4 - \sqrt{5}\)[/tex], we need to find the other zeros.
First, recall that for a polynomial with rational coefficients, complex zeros and irrational zeros must occur in conjugate pairs. This implies that if one zero is [tex]\(i\)[/tex] or [tex]\(a + b\sqrt{c}\)[/tex], then the corresponding zero must be [tex]\(-i\)[/tex] or [tex]\(a - b\sqrt{c}\)[/tex], respectively.
With this in mind, here are the given and implied zeros:
- Given zero: [tex]\(i\)[/tex]
- Its conjugate: [tex]\(-i\)[/tex]
- Given zero: [tex]\(4 - \sqrt{5}\)[/tex]
- Its conjugate: [tex]\(4 + \sqrt{5}\)[/tex]
Therefore, the complete list of zeros for the polynomial is:
[tex]$i, -i, 4-\sqrt{5}, 4+\sqrt{5}$[/tex]
The other zeros are:
[tex]$-i, 4 + \sqrt{5}$[/tex]
