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Question: Why does simplify produce wrong results for this polynomial?

Question:Why does simplify produce wrong results for this polynomial?

ispr 10 Maple 15
polynomial simplify
August 12 2013
1

When defining a polynomial as follows

p:=x->x^(r+1)-(r+1)*Sum((-1)^(r+k)*(r+k)!*(r+1)!*r!/((k!)^2*(r-k+1)!*(2*r+1)!)*x^k,k=0..r);

the result of

simplify((eval(p(1),r=1)));

is given as 1 whereas hand evaluation and also plotting the polynomial clearly shows that the result is 0. How can this behaviour be explained?
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