To determine the other zeros of a polynomial of degree 5 with rational coefficients, we need to consider the property that such polynomials have conjugate pairs as zeros when non-rational or complex numbers are involved. Below are the steps to find the other zeros given the provided zeros:
1. Identify the given zeros:
- [tex]\( -\frac{1}{6} \)[/tex]
- [tex]\( \sqrt{7} \)[/tex]
- [tex]\( -4i \)[/tex]
2. Apply the property of rational coefficients:
For a polynomial with rational coefficients, any irrational or complex zeros must occur in conjugate pairs. This means:
- If [tex]\( \sqrt{7} \)[/tex] is a zero, then [tex]\( -\sqrt{7} \)[/tex] must also be a zero.
- If [tex]\( -4i \)[/tex] is a zero, then [tex]\( 4i \)[/tex] must also be a zero.
3. Determine the missing conjugate pairs:
- Since [tex]\( \sqrt{7} \)[/tex] is a zero, its conjugate [tex]\( -\sqrt{7} \)[/tex] is another zero.
- Since [tex]\( -4i \)[/tex] is a zero, its conjugate [tex]\( 4i \)[/tex] is another zero.
Therefore, the additional zeros needed to complete the polynomial of degree 5 are:
[tex]\[ -\sqrt{7}, 4i \][/tex]
Hence, the other zero(s) is/are:
[tex]\[ -\sqrt{7}, 4i \][/tex]
