Question
The polynomial a8x8 +a7x7 + … + a0 has a8 = 1, a7 = -4, a6 = 7 and all its roots positive and real. Find the possible values for a0.
Answer
Let the roots be xi. We have Sum xi2 = 42 – 2·7 = 2. By Cauchy-Schwarz inequality, we have (x1·1 + … + x8·1) ≤ (x12 + … + x82)1/2(12 + … + 12)1/2 with equality iff all xi are equal. Hence all xi are equal. So they must all be 1/2.
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