To solve for the other zeros of a polynomial function of degree 4 with rational coefficients, given the zeros [tex]\(-i\)[/tex] and [tex]\(4 - \sqrt{5}\)[/tex], we utilize the properties of polynomials with rational coefficients.
A polynomial with rational coefficients must have zeros that occur in conjugate pairs if those zeros are irrational or complex. This implies that:
1. If [tex]\( -i \)[/tex] is a zero, its complex conjugate [tex]\( i \)[/tex] must also be a zero.
2. If [tex]\( 4 - \sqrt{5} \)[/tex] is a zero, its conjugate [tex]\( 4 + \sqrt{5} \)[/tex] must also be a zero.
Given [tex]\(-i\)[/tex] and [tex]\(4 - \sqrt{5}\)[/tex] as zeros, we therefore identify the following zeros as well:
- The complex conjugate of [tex]\(-i\)[/tex] is [tex]\(i\)[/tex].
- The conjugate of [tex]\(4 - \sqrt{5}\)[/tex] is [tex]\(4 + \sqrt{5}\)[/tex].
Thus, the zeros of the polynomial are:
- [tex]\(-i\)[/tex]
- [tex]\(i\)[/tex]
- [tex]\(4 - \sqrt{5}\)[/tex]
- [tex]\(4 + \sqrt{5}\)[/tex]
Therefore, the other zeros of the polynomial, in addition to [tex]\(-i\)[/tex] and [tex]\(4 - \sqrt{5}\)[/tex], are [tex]\(i\)[/tex] and [tex]\(4 + \sqrt{5}\)[/tex].
