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if a polynomial equation has the solution 2 + 3i, its conjugate, 2 - 3i, must also be a solution. true or false
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Answer: It depends.
- True if the polynomial has real coefficients. By the Complex Conjugate Root Theorem, nonreal complex roots occur in conjugate pairs: if p(x) has real coefficients and p(2+3i)=0 then p(2−3i)=conj(p(2+3i))=0.
- False in general if coefficients may be complex. Counterexample: p(x)=x−(2+3i) has root 2+3i but not 2−3i.
- True if the polynomial has real coefficients. By the Complex Conjugate Root Theorem, nonreal complex roots occur in conjugate pairs: if p(x) has real coefficients and p(2+3i)=0 then p(2−3i)=conj(p(2+3i))=0.
- False in general if coefficients may be complex. Counterexample: p(x)=x−(2+3i) has root 2+3i but not 2−3i.
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