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Question Number 190536 by cortano12 last updated on 05/Apr/23

$$\mathrm{If}\mathrm{p},\mathrm{q}\mathrm{and}\mathrm{r}\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{equation}$$$${\mathrm{x}}^3-{3\mathrm{x}}^2+1=0\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}$$$$\mathrm{of}\sqrt[3]{3\mathrm{p}-2}+\sqrt[3]{3\mathrm{q}-2}+\sqrt[3]{3\mathrm{r}-2}$$

Answered by Frix last updated on 05/Apr/23

$$t=\sqrt[3]{3x-2}\iff x=\frac{t^3+2}{3}$$$$\mathrm{Inserting},\mathrm{transforming}$$$$t^9-3t^6-24t^3-1=0$$$$\mathrm{Searching}\mathrm{for}\mathrm{factors}\left(\mathrm{using}\mathrm{software}\right)\Rightarrow$$$$t^3+0t^2-3t-1\left[\mathrm{the}\mathrm{other}\mathrm{factor}\mathrm{has}\mathrm{no}\mathrm{real}\mathrm{roots}\right]$$$$\Rightarrow \mathrm{answer}\mathrm{is}0$$

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