Question
Suppose that a polynomial function of degree 4 with rational coefficients has the given numbers as zeros. Find the other zero(s). \( 6 i, 0,-8 \)
Ask by Horton Jimenez. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The other zero is \( -6i \).
Solution
To find the other zero(s) of the polynomial function of degree 4 with rational coefficients, we need to consider the properties of polynomials with rational coefficients. Specifically, if a polynomial has complex roots, they must occur in conjugate pairs.
Given the zeros:
- \( 6i \) (a complex zero)
- \( 0 \) (a real zero)
- \( -8 \) (a real zero)
Since \( 6i \) is a zero, its conjugate \( -6i \) must also be a zero of the polynomial.
Thus, the complete set of zeros for the polynomial is:
- \( 6i \)
- \( -6i \)
- \( 0 \)
- \( -8 \)
Therefore, the other zero is \( -6i \).
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Simplify this solution Extra Insights
In the realm of polynomials, if a polynomial has real or rational coefficients, complex roots must appear in conjugate pairs. Since \(6i\) is a zero, its conjugate, \(-6i\), is also a zero of the polynomial. Thus, the zeros of the polynomial are \(6i\), \(-6i\), \(0\), and \(-8\). Combining these, our complete set of zeros reveals the polynomial is guaranteed to be a degree 4 polynomial, as it has exactly four roots! So, you have \(6i, -6i, 0, -8\) as your actual zeros. Happy fact-finding!
