Question
Suppose a polynomial function of degree 4 with rational coefficients has the following given numbers as zeros. Find the other zero(s).
The other zero(s) is/are
(Type an exact answer, using radicals and
The other zero(s) is/are
(Type an exact answer, using radicals and
Ask by Craig Mann. in the United States
Mar 16,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The other zeros are
Solution
To find the other zeros of the polynomial function of degree 4 with rational coefficients, we can use the following properties of polynomials:
- If a polynomial has rational coefficients, then any non-real complex zeros must occur in conjugate pairs.
- If a polynomial has rational coefficients, then any irrational zeros must also occur in conjugate pairs.
Given the zeros:
-
(a complex zero), its conjugate must also be a zero. -
(an irrational zero), its conjugate must also be a zero.
Thus, the other zeros of the polynomial are:
Therefore, the other zero(s) is/are:
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Simplify this solution Beyond the Answer
Since the polynomial has rational coefficients, the complex conjugate of any imaginary zero must also be a zero. Therefore, if
Additionally,
Thus, the other zeros are:
