A non-zero polynomial f(x) of degree 3 has roots at x = -1, x = 2 and x = 3. Which of the following must be TRUE? A. f(0)*f(-2) < 0 B. f(0)*f(-2) > 0 C. f(0)*f(4) < 0 D. f(0)*f(4) > 0

GATE 2014 · Discrete Mathematics · Polynomials · medium

Answer: f(0)*f(-2) = -120a^2 < 0 always, so option A must be TRUE.

1. Write f in factored form: Since f has degree 3 and roots -1, 2, 3, write f(x) = a(x+1)(x-2)(x-3) for some nonzero constant a.
2. Evaluate at the two test points: f(0) = a*(1)*(-2)*(-3) = 6a. f(-2) = a*(-2+1)*(-2-2)*(-2-3) = a*(-1)*(-4)*(-5) = -20a. Note f(0) and f(-2) have opposite signs.
3. Compute the product: f(0)*f(-2) = 6a*(-20a) = -120a^2. Since a != 0, we have a^2 > 0, so -120a^2 < 0. This is ALWAYS negative.