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Q-If-is-a-root-of-the-equation-x-3-3x-1-0-prove-that-the-other-roots-are-2-2-and-2-2-Please-help-

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Question Number 33193 by SammyKT last updated on 12/Apr/18
Q. If α is a root of the equation x^3 −3x−1=0, prove that the other roots are 2−α^2 and α^2 −α−2. Please help.
$$\mathrm{Q}.\:\:\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{\mathrm{3}} −\mathrm{3}{x}−\mathrm{1}=\mathrm{0}, \\ $$$$\:\:\:\:\:\:\:\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{other}\:\mathrm{roots}\:\mathrm{are} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{2}−\alpha^{\mathrm{2}} \:\mathrm{and}\:\alpha^{\mathrm{2}} −\alpha−\mathrm{2}. \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Please}\:\mathrm{help}. \\ $$
Answered by MJS last updated on 12/Apr/18
x^3 +0x^2 −3x−1=0 (x−α)(x−β)(x−γ)=0 x^3 −(α+β+γ)x^2 +(αβ+αγ+βγ)x−αβγ=0 ⇒ −(α+β+γ)=0 now we set β=2−α^2 and γ=α^2 −α−2 α+2−α^2 +α^2 −α−2=0 true
$${x}^{\mathrm{3}} +\mathrm{0}{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{1}=\mathrm{0} \\ $$$$ \\ $$$$\left({x}−\alpha\right)\left({x}−\beta\right)\left({x}−\gamma\right)=\mathrm{0} \\ $$$${x}^{\mathrm{3}} −\left(\alpha+\beta+\gamma\right){x}^{\mathrm{2}} +\left(\alpha\beta+\alpha\gamma+\beta\gamma\right){x}−\alpha\beta\gamma=\mathrm{0} \\ $$$$\Rightarrow\:−\left(\alpha+\beta+\gamma\right)=\mathrm{0} \\ $$$$ \\ $$$$\mathrm{now}\:\mathrm{we}\:\mathrm{set}\:\beta=\mathrm{2}−\alpha^{\mathrm{2}} \:\mathrm{and}\:\gamma=\alpha^{\mathrm{2}} −\alpha−\mathrm{2} \\ $$$$ \\ $$$$\alpha+\mathrm{2}−\alpha^{\mathrm{2}} +\alpha^{\mathrm{2}} −\alpha−\mathrm{2}=\mathrm{0} \\ $$$$\mathrm{true} \\ $$
Commented by SammyKT last updated on 13/Apr/18
Thank you
$$\mathrm{Thank}\:\mathrm{you} \\ $$

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