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Let f(x) be a polynomial of degree 5 with leading coefficient 1 such that f(1) = 5, f(2) = 4, f(3) = 3, f(4) = 2, f(5) = 1. Find f(6).

1. 0
2. 24
3. 120
4. 720

Correct answer: 120

Solution

g(x) = f(x)-(6-x) has roots 1,2,3,4,5 and is degree 5 monic, so g(x)=(x-1)(x-2)(x-3)(x-4)(x-5). f(6)=g(6)+(6-6)=5!=120.

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