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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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GPT-5 mini
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Not enough information as stated.
- If the polynomial has real coefficients, then True. Nonreal roots occur in complex-conjugate pairs: if p(2+3i)=0 and all coefficients of p are real, then taking complex conjugates gives p(2-3i)=0.
- If the coefficients may be complex, False. Example: p(x) = x - (2+3i) has 2+3i as a root but 2-3i is not a root.
- If the polynomial has real coefficients, then True. Nonreal roots occur in complex-conjugate pairs: if p(2+3i)=0 and all coefficients of p are real, then taking complex conjugates gives p(2-3i)=0.
- If the coefficients may be complex, False. Example: p(x) = x - (2+3i) has 2+3i as a root but 2-3i is not a root.
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