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Question Number 142232 by liberty last updated on 28/May/21

Answered by MJS_new last updated on 28/May/21

$$1.x=0$$$$\Rightarrow y^3=1\wedge y^4-y=0\Rightarrow y=1\vee y=-\frac{1}{2}\pm \frac{\sqrt{3}}{2}\mathrm{i}$$$$2.y=0...\Rightarrow x=1\vee x=-\frac{1}{2}\pm \frac{\sqrt{3}}{2}\mathrm{i}$$$$3.y=x...\Rightarrow x=y=1\vee x=y=-\frac{1}{2}\pm \frac{\sqrt{3}}{2}\mathrm{i}$$$$4.y=px\wedge x\ne 0\wedge p\ne 0\wedge p\ne 1$$$$(p^3-p^2+1)x^3=1\Rightarrow x^3=\frac{1}{p^3-p^2+1}$$$$(4-p^4)x^4=(4-p)x\Rightarrow x^3=\frac{4-p}{4-p^4}$$$$p^3-p^2+1=\frac{4-p^4}{4-p}$$$$5p^3-4p^2-p=0$$$$p=-\frac{1}{5}\Rightarrow x=\frac{5}{\sqrt[3]{119}}\wedge y=-\frac{1}{\sqrt[3]{119}}\vee x=\frac{5}{\sqrt[3]{119}}(-\frac{1}{2}\pm \frac{\sqrt{3}}{2}\mathrm{i})\wedge y=-\frac{1}{\sqrt[3]{119}}(-\frac{1}{2}\pm \frac{\sqrt{3}}{2}\mathrm{i})$$

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