4. Suppose n is a positive odd integer. Prove that f(x) in Q[x]. (Hint. Assume the contrary and first reduce it to the case where f(x) = g(x)h(x) for some non-constant integer polynomials g(x) and h(x). Then consider f (i) for integer i in [1, n], and think about g(x)² – 1 and h(x)² – 1.) (x – 1)(x – 2) ·. (x – n) – 1 is irreducible

4. Suppose n is a positive odd integer. Prove that f(x) in Q[x]. (Hint. Assume the contrary and first reduce it to the case where f(x) = g(x)h(x) for some non-constant integer polynomials g(x) and h(x). Then consider f (i) for integer i in [1, n], and think about g(x)² – 1 and h(x)² – 1.) (x – 1)(x – 2) ·. (x – n) – 1 is irreducible

Name: 4. Suppose n is a positive odd integer. Prove that f(x) = (x – 1)(x –
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Description: 4. Suppose n is a positive odd integer. Prove that f(x) = (x – 1)(x – 2) . .· (x – n) – 1 is irreduciblein Q[x). (Hint. Assume the contrary and first reduce it to the case where f(x) = g(x)h(x) for somenon-constant integer polynomials g(x) and h(x).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Suppose n is a positive odd integer. Prove that f(x) = (x – 1)(x – 2) . .· (x – n) – 1 is irreducible
in Q[x). (Hint. Assume the contrary and first reduce it to the case where f(x) = g(x)h(x) for some
non-constant integer polynomials g(x) and h(x). Then consider f(i) for integer i in [1, n], and think
about g(x)² – 1 and h(x)² – 1.)

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