To determine the other zero of the polynomial function with the given zeros [tex]\(7i, 0, -1\)[/tex], and rational coefficients, follow these steps:
1. Identify Given Zeros:
The given zeros are [tex]\(7i\)[/tex], [tex]\(0\)[/tex], and [tex]\(-1\)[/tex].
2. Understand Polynomial Conjugate Pairs:
For a polynomial with rational coefficients, any complex zeros must come in conjugate pairs. This means if [tex]\(7i\)[/tex] is a zero, its complex conjugate [tex]\(-7i\)[/tex] must also be a zero.
3. List All Zeros:
Given the conjugate pair rule, the zeros of the polynomial are:
- [tex]\(0\)[/tex]
- [tex]\( -1\)[/tex]
- [tex]\( 7i\)[/tex]
- [tex]\( -7i\)[/tex]
4. Conclusion:
The other zero of the polynomial, which ensures the polynomial has rational coefficients, is [tex]\(-7i\)[/tex].
So, the complete list of zeros for this polynomial is:
- [tex]\(0\)[/tex]
- [tex]\(-1\)[/tex]
- [tex]\(7i\)[/tex]
- [tex]\(-7i\)[/tex]
Thus, the polynomial has the zeros [tex]\(0, -1, 7i, -7i\)[/tex], and the other zero you were looking for is [tex]\( \boxed{-7i} \)[/tex].
