Let f(x) = x^3 - qx , where q 0 . Let a₁, a₂, a₃, a₄, a₅ be in an arithmetic progression with mean 0 and an integer common difference d 0 . If |f(a_i)| = 16 for i 1, 2, 4, 5 , then the square of the difference between the largest and sm

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Let f(x) = x^3 - qx , where q > 0 . Let a₁, a₂, a₃, a₄, a₅ be in an arithmetic progression with mean 0 and an integer common difference d > 0 . If |f(a_i)| = 16 for i 1, 2, 4, 5 , then the square of the difference between the largest and smallest roots of f(x) = 0 is

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Correct answer: 48

Since the arithmetic progression has mean 0 and common difference d , the five terms are -2d, -d, 0, d, 2d . Thus, a₁ = -2d , a₂ = -d , a₄ = d , and a₅ = 2d . We are given |f(d)| = 16 and |f(2d)| = 16 . For x = d : |d^3 - qd| = 16 For x = 2d : |8d^3 - 2qd| = 16 This implies 8d^3 - 2qd = (d^3 - qd) . Case 1: 8d^3 - 2qd = d^3 - qd 7d^3 = qd q = 7d^2 (since d > 0 ) Substitute q = 7d^2 into |d^3 - qd| = 16 : |d^3 - 7d^3| = 16 |-6d^3| = 16 6d^3 = 16 d^3 = 8 3 . This does not yield an integer value for d , so we reject t

Let f(x) = x^3 - qx , where q > 0 . Let a₁, a₂, a₃, a₄, a₅ be in an arithmetic progression with mean 0 and an integer common difference d > 0 . If |f(a_i)| = 16 for i 1, 2, 4, 5 , then the square of the difference between the largest and smallest roots of f(x) = 0 is

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