InterviewSolution
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InterviewSolution
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Let f(x) = x3 + ax2 + bx + c and g(x) = x3 + bx2 + cx + a, where a; b; c are integers with c ≠ 0. Suppose that the following conditions hold: (a) f(1) = 0; (b) the roots of g(x) are squares of the roots of f(x). Find the value of a2013 + b2013 + c2013. |
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Answer» Note that g(1) = f(1) = 0, so 1 is a root of both f(x) and g(x). Let p and q be the other two roots of f(x), so p2 and q2 are the other two roots of g(x). We then get pq = - c and p2q2 = -a, so a = - c2. Also, (- a)2 = (p+q+1)2 = p2+q2+1+2(pq+p+q) = - b+2b = b. Therefore b = c4. Since f(1) = 0 we therefore get 1 + c - c2 + c4 = 0. Factorising, we get (c + 1)(c3 - c2 + 1) = 0. Note that c3 - c2 + 1 = 0 has no integer root and hence c = -1; b = 1; a = -1. Therefore a2013 + b2013 + c2013 = -1. |
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