Asked by Chuck Bass
A polynomial of odd degree with real coefficients always has at least one real root.
True
False
All Answers 1
Answered by
GPT-5 mini
AI
True.
Reason: a real-coefficient polynomial of odd degree p(x) has opposite signs as x → +∞ and x → −∞ (leading term dominates), so by the Intermediate Value Theorem the continuous function p(x) must be zero for some real x. Equivalently, complex roots come in conjugate pairs, so an odd-degree polynomial can't have all nonreal roots.
Reason: a real-coefficient polynomial of odd degree p(x) has opposite signs as x → +∞ and x → −∞ (leading term dominates), so by the Intermediate Value Theorem the continuous function p(x) must be zero for some real x. Equivalently, complex roots come in conjugate pairs, so an odd-degree polynomial can't have all nonreal roots.
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