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Question-193087

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Question Number 193087 by Abdullahrussell last updated on 03/Jun/23
Answered by Frix last updated on 03/Jun/23
x^4 +2x^3 −4x−2 (x+(1/2))^4 −(3/2)(x+(1/2))^2 −3(x+(1/2))−(3/(16)) ((x+(1/2))^2 −a(x+(1/2))−b)((x+(1/2))^2 +a(x+(1/2))−c) Comparing constants ⇒ ((x+(1/2))^2 −(√3)(x+(1/2))+(3/4)−((√3)/2))((x+(1/2))^2 +(√3)(x+(1/2))+(3/4)+((√3)/2)) ⇔ (x^2 +(1−(√3))x+1−(√3))(x^2 +(1+(√3))x+1+(√3)) This is your (x^2 +px+q)(x^2 +rx+s) (x+((1−(√3)−((12))^(1/4) )/2))(x+((1−(√3)+((12))^(1/4) )/2))(x+((1+(√3))/2)+(((12))^(1/4) /2)i)(x+((1+(√3))/2)−(((12))^(1/4) /2)i) This is your (x+t)(x+u)(+v)(x+w)
$${x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{3}} −\mathrm{4}{x}−\mathrm{2} \\ $$$$\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{4}} −\frac{\mathrm{3}}{\mathrm{2}}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{3}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\frac{\mathrm{3}}{\mathrm{16}} \\ $$$$\left(\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} −{a}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)−{b}\right)\left(\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +{a}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)−{c}\right) \\ $$$$\mathrm{Comparing}\:\mathrm{constants}\:\Rightarrow \\ $$$$\left(\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} −\sqrt{\mathrm{3}}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{3}}{\mathrm{4}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)\left(\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\sqrt{\mathrm{3}}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{3}}{\mathrm{4}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right) \\ $$$$\Leftrightarrow \\ $$$$\left({x}^{\mathrm{2}} +\left(\mathrm{1}−\sqrt{\mathrm{3}}\right){x}+\mathrm{1}−\sqrt{\mathrm{3}}\right)\left({x}^{\mathrm{2}} +\left(\mathrm{1}+\sqrt{\mathrm{3}}\right){x}+\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\:\:\:\:\:\mathrm{This}\:\mathrm{is}\:\mathrm{your}\:\left({x}^{\mathrm{2}} +{px}+{q}\right)\left({x}^{\mathrm{2}} +{rx}+{s}\right) \\ $$$$\left({x}+\frac{\mathrm{1}−\sqrt{\mathrm{3}}−\sqrt[{\mathrm{4}}]{\mathrm{12}}}{\mathrm{2}}\right)\left({x}+\frac{\mathrm{1}−\sqrt{\mathrm{3}}+\sqrt[{\mathrm{4}}]{\mathrm{12}}}{\mathrm{2}}\right)\left({x}+\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}+\frac{\sqrt[{\mathrm{4}}]{\mathrm{12}}}{\mathrm{2}}\mathrm{i}\right)\left({x}+\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}−\frac{\sqrt[{\mathrm{4}}]{\mathrm{12}}}{\mathrm{2}}\mathrm{i}\right) \\ $$$$\:\:\:\:\:\mathrm{This}\:\mathrm{is}\:\mathrm{your}\:\left({x}+{t}\right)\left({x}+{u}\right)\left(+{v}\right)\left({x}+{w}\right) \\ $$

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