1985 IMO SL #11

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+  a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.



Let $-x_1, \cdots, -x_6$ be the roots of $P(x)$. Let $s_{k,i}$, where $k \le i \le 6$, denote the sum of all products of $k$ of the numbers $x_1, \cdots, x_i$. Using Vieta's Formulas, note that $a_k=s_{k,6}$. Also, notice that $$s_{k,i}=s_{k-1,i-1} x_i+s_{k,i-1}.$$ Using this, we can compute all $a_k$ as shown in the below diagram, where horizontal arrow represents multiplication and vertical arrow represents addition. x 1 s 2 , 2 s 3 , 3 s 4 , 4 s 5 , 5 a 6 s 1 , 2 s 2 , 3 s 3 , 4 s 4 , 5 a 5 s 1 , 3 s 2 , 4 s 3 , 5 a 4 s 1 , 4 s 2 , 5 a 3 s 1 , 5 a 2 a 1 We are done. $\square$  







Comments

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    ReplyDelete
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