Let a,b,c,d be real numbers such that a+b+c+da+2b+4c+8da−5b+25c−125da+6b+36c+216d=1,=16,=625,=1296.Enter the ordered quadruple (a,b,c,d).
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Consider the polynomial p(x)=x4−dx3−cx2−bx−a.Then p(1)=1−d−c−b−a=0. Similarly, p(2)p(−5)p(6)=16−8d−4c−2b−a=0,=625−125d−25c−5b−a=0,=1296−216d−36c−6b−a=0.Since p(x) has degree 4 and is monic, p(x)=(x−1)(x−2)(x+5)(x−6)=x4−4x3−25x2+88x−60.Hence, (a,b,c,d)=(60,−88,25,4).