Question
Suppose that a polynomial function of degree 5 with rational coefficients has the given numbers as zeros. Find the other zero(s). \( 7,-5+2 i, 5-\sqrt{3} \) The other zero(s) is/are (Type an exact answer, using radicals and \( i \) as needed. Use a comma to separate answers as needed.)
Ask by King Wang. in the United States
Mar 16,2025
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Answer
The other zeros are \( -5-2i \) and \( 5+\sqrt{3} \).
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For a polynomial with rational coefficients, any non-real or irrational zeros must occur in conjugate pairs. Since \( -5 + 2i \) is a zero, its conjugate \( -5 - 2i \) must also be a zero. Similarly, \( 5 - \sqrt{3} \) implies the zero \( 5 + \sqrt{3} \) is also present. Therefore, the other zeros are \( -5 - 2i \) and \( 5 + \sqrt{3} \). The other zero(s) is/are \( -5 - 2i, 5 + \sqrt{3} \).