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Let f(x) be a polynomial of degree 3 such that f(k) = -2/k for k = 2, 3, 4, 5. Find the value of 52 - 10 f(10).

1. 26
2. 22
3. 30
4. 28

Correct answer: 26

Solution

Define g(x) = x*f(x) + 2. Since f is degree 3, g is degree 4. At x = 2,3,4,5: g(k) = k*f(k)+2 = k*(-2/k)+2 = -2+2 = 0. So g has four roots: x=2,3,4,5. Thus g(x) = A(x-2)(x-3)(x-4)(x-5). At x=0: g(0) = 0*f(0)+2 = 2, and A*(-2)(-3)(-4)(-5) = 120A = 2, giving A = 1/60. Then g(10) = (1/60)*(8)*(7)*(6)*(5) = 1680/60 = 28. So 10*f(10) = g(10)-2 = 26. Finally 52 - 10*f(10) = 52 - 26 = 26.

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