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 October 31st, 2015, 06:19 AM #1 Newbie   Joined: Oct 2015 From: Edinburgh Posts: 6 Thanks: 0 Polynomial problem Can you help me with this problem please: Find all pairs of polynomials A, B with real coefficients that for each real number expressions A(x^2 + 1) = B(x)^2 + 2x B(x^2 + 1) = A(x)^2 are true. November 2nd, 2015, 07:38 AM #2 Newbie   Joined: Oct 2015 From: Edinburgh Posts: 6 Thanks: 0 Any ideas? November 2nd, 2015, 09:25 AM #3 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms The second line implies that the degree of A is the same as the degree of B. B(x^2 + 1) has only terms with even powers, so the powers of terms in A(x) with nonzero coefficients must either be all even or all odd. A(x^2 + 1) has only terms with even powers, so the powers of terms in B(x)^2 + 2x with nonzero coefficients must either be all even or all odd. Let B(x) = a + bx + O(x^2), then B(x)^2 = a^2 + 2abx + O(x^2) and so B(x)^2 + 2x = a^2 + (2ab + 2)x + O(x^2) and so either a or 2ab + 2 (or both) must be 0. If a is nonzero, that is, if B(x) is not a multiple of x, then b = -1/a. Tags polynomial, polynomials, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post taytay22 Pre-Calculus 2 August 22nd, 2015 12:13 PM DapperDub Pre-Calculus 3 August 21st, 2015 07:48 AM KyVanchhay Algebra 2 June 27th, 2013 01:07 PM nukem4111 Algebra 0 April 13th, 2013 11:34 AM Pell's fish Number Theory 5 August 30th, 2010 03:04 PM

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