### 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. $$\begin{array}{ccccccccccc}{x}_{1}& \to & {s}_{2,2}& \to & {s}_{3,3}& \to & {s}_{4,4}& \to & {s}_{5,5}& \to & {a}_{6}\\ \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ {s}_{1,2}& \to & {s}_{2,3}& \to & {s}_{3,4}& \to & {s}_{4,5}& \to & {a}_{5}\\ \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ {s}_{1,3}& \to & {s}_{2,4}& \to & {s}_{3,5}& \to & {a}_{4}& & & \\ \downarrow & & \downarrow & & \downarrow & & & & \\ {s}_{1,4}& \to & {s}_{2,5}& \to & {a}_{3}& & & & \\ \downarrow & & \downarrow & & & & & \\ {s}_{1,5}& \to & {a}_{2}& & & & & \\ \downarrow & & & & & & & \\ {a}_{1}& & & & & \end{array}$$ We are done. $\square$

beautifully explained

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आप बहुत प्रो हैं।