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A non-zero polynomial f(x) of degree 3 has roots at x = 1, x = 2 and x = 3. Which one of the following must be TRUE?

1. f(0)f(4) < 0
2. f(0)f(4) > 0
3. f(0) + f(4) > 0
4. f(0) + f(4) < 0

Correct answer: f(0)f(4) < 0

Solution

Write f(x)=a(x-1)(x-2)(x-3) with a nonzero. Then f(0)=a(-1)(-2)(-3)=-6a and f(4)=a(3)(2)(1)=6a, so f(0)f(4)=-36a^2, which is strictly negative for every nonzero a. Hence the statement that must be true is f(0)f(4)<0 (index 0), not f(0)f(4)>0.

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