Let $a,b$ and $c$ be the three real roots of the polynomial $x^3-5x^2+5x+1=0$ find the value of $(a^2+ab+b^2)(b^2+bc+c^2)(c^2+ca+a^2)$
2026-04-08 02:26:31.1775615191
Polynomial problem TUGMOs Grade9
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Let $f(x)=x^3-5x^2+5x+1$, and note that any solution must satisfy the equation $$x^3=5x^2-5x-1 \tag{1}$$
Note $$\begin{align} \prod_{cyc}(a^2+ab+b^2) &=\prod_{cyc}\dfrac{a^3-b^3}{a-b} \\ &=\prod_{cyc} \dfrac{5a^2-5a-5b^2+5b}{a-b} \quad (\because (1))\\ &= \prod_{cyc} (5a+5b-5) \\ &= 125 \prod_{cyc} (4-c) \quad (\because a+b-1=4-c) \\ &= 125 f(4) =625\end{align}$$