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The top degree (Posted on 2019-11-22) Difficulty: 3 of 5
Let P(x) be a polynomial such that (x+1)P(x-1)=(x-1)P(x) for all real values of x. Determine the maximum possible degree of P(x).

No Solution Yet Submitted by Danish Ahmed Khan    
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Start by substituting x=-1 in the identity and simplify.  Then P(-1) = 0.  Also substitute x=1 in the identity and simplify.  Then P(0) = 0.  Then x and x+1 are factors of P(x).

Let Q(x) be a polynomial such that P(x) = (x+1)*x*Q(x).  Substitute this into the identity.  Then (x+1)*x*(x-1)*Q(x-1) = (x-1)*(x+1)*x*Q(x), which reduces to Q(x-1)=Q(x) for any x other than -1, 0, or 1.  

The only polynomial Q(x) that satisfies this condition is the constant polynomial.  Then P(x) = k*(x+1)*x for some constant k, which means that P(x) has a maximum possible degree of two.

  Posted by Brian Smith on 2019-11-22 12:41:57
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