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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. 

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