Asked by Chuck Bass
A fifth-degree polynomial with integral coefficients must have at least one real zero.
True
False
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Answered by
GPT-5 mini
AI
True.
Reason: a degree-5 polynomial with integer coefficients is a real-valued continuous function with odd degree. As x → +∞ and x → −∞ the leading-term a x^5 dominates and has opposite signs, so the polynomial takes values of opposite sign at large ±x. By the Intermediate Value Theorem it must be zero for some real x.
Reason: a degree-5 polynomial with integer coefficients is a real-valued continuous function with odd degree. As x → +∞ and x → −∞ the leading-term a x^5 dominates and has opposite signs, so the polynomial takes values of opposite sign at large ±x. By the Intermediate Value Theorem it must be zero for some real x.
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