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Question Number 212175 by RojaTaniya last updated on 05/Oct/24
Answered by A5T last updated on 05/Oct/24
Let a+b+c=x;ab+bc+ca=y;abc=z a^2 +b^2 +c^2 =x^2 −2y=20...(i) a^3 +b^3 +c^3 =x(x^2 −3y)+3z=x^3 −3xy+3z=44...(ii) (a+b)(b+c)(c+a)=xy−z=156..(iii) (ii)+3(iii): x^3 =512⇒x=8⇒y=22⇒z=20 ⇒a,b,c roots of: s^3 −8s^2 +22s−20=0 ⇒(s−2)(s^2 −6s+10)=0 ⇒(a,b,c)=(2,3+i,3−i) upto permutation ⇒a^6 +b^6 +c^6 =2^6 +(3+i)^6 +(3−i)^6 =−640
$${Let}\:{a}+{b}+{c}={x};{ab}+{bc}+{ca}={y};{abc}={z} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} ={x}^{\mathrm{2}} −\mathrm{2}{y}=\mathrm{20}…\left({i}\right) \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} ={x}\left({x}^{\mathrm{2}} −\mathrm{3}{y}\right)+\mathrm{3}{z}={x}^{\mathrm{3}} −\mathrm{3}{xy}+\mathrm{3}{z}=\mathrm{44}…\left({ii}\right) \\ $$$$\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right)={xy}−{z}=\mathrm{156}..\left({iii}\right) \\ $$$$\left({ii}\right)+\mathrm{3}\left({iii}\right):\:{x}^{\mathrm{3}} =\mathrm{512}\Rightarrow{x}=\mathrm{8}\Rightarrow{y}=\mathrm{22}\Rightarrow{z}=\mathrm{20} \\ $$$$\Rightarrow{a},{b},{c}\:{roots}\:{of}:\:{s}^{\mathrm{3}} −\mathrm{8}{s}^{\mathrm{2}} +\mathrm{22}{s}−\mathrm{20}=\mathrm{0} \\ $$$$\Rightarrow\left({s}−\mathrm{2}\right)\left({s}^{\mathrm{2}} −\mathrm{6}{s}+\mathrm{10}\right)=\mathrm{0} \\ $$$$\Rightarrow\left({a},{b},{c}\right)=\left(\mathrm{2},\mathrm{3}+{i},\mathrm{3}−{i}\right)\:{upto}\:{permutation} \\ $$$$\Rightarrow{a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =\mathrm{2}^{\mathrm{6}} +\left(\mathrm{3}+{i}\right)^{\mathrm{6}} +\left(\mathrm{3}−{i}\right)^{\mathrm{6}} =−\mathrm{640} \\ $$
Commented by A5T last updated on 05/Oct/24
a^6 +b^6 +c^6 =(a^2 )^3 +(b^2 )^3 +(c^2 )^3 =(a^2 +b^2 +c^2 )[(a^2 +b^2 +c^2 )^2 −3a^2 b^2 −3b^2 c^2 −3c^2 a^2 ]+3a^2 b^2 c^2 =20[400−3(ab+bc+ca)^2 +6abc(a+b+c)]+3z^2 =20[400−3(22)^2 +3(20)(8)]+3(20)^2 =−640
$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =\left({a}^{\mathrm{2}} \right)^{\mathrm{3}} +\left({b}^{\mathrm{2}} \right)^{\mathrm{3}} +\left({c}^{\mathrm{2}} \right)^{\mathrm{3}} \\ $$$$=\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)\left[\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)^{\mathrm{2}} −\mathrm{3}{a}^{\mathrm{2}} {b}^{\mathrm{2}} −\mathrm{3}{b}^{\mathrm{2}} {c}^{\mathrm{2}} −\mathrm{3}{c}^{\mathrm{2}} {a}^{\mathrm{2}} \right]+\mathrm{3}{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} \\ $$$$=\mathrm{20}\left[\mathrm{400}−\mathrm{3}\left({ab}+{bc}+{ca}\right)^{\mathrm{2}} +\mathrm{6}{abc}\left({a}+{b}+{c}\right)\right]+\mathrm{3}{z}^{\mathrm{2}} \\ $$$$=\mathrm{20}\left[\mathrm{400}−\mathrm{3}\left(\mathrm{22}\right)^{\mathrm{2}} +\mathrm{3}\left(\mathrm{20}\right)\left(\mathrm{8}\right)\right]+\mathrm{3}\left(\mathrm{20}\right)^{\mathrm{2}} =−\mathrm{640} \\ $$
Commented by A5T last updated on 05/Oct/24
a^6 +b^6 +c^6 =(a^3 )^2 +(b^3 )^2 +(c^3 )^2 =(a^3 +b^3 +c^3 )^2 −2(a^3 b^3 +b^3 c^3 +c^3 a^3 ) =44^2 −2[(ab+bc+ca)^3 −3abc(ab+bc+ca)(a+b+c) +3(abc)^2 ] =44^2 −2[22^3 −3(20)(22)(8)+3(20)^2 ]=−640
$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =\left({a}^{\mathrm{3}} \right)^{\mathrm{2}} +\left({b}^{\mathrm{3}} \right)^{\mathrm{2}} +\left({c}^{\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$=\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} \right)^{\mathrm{2}} −\mathrm{2}\left({a}^{\mathrm{3}} {b}^{\mathrm{3}} +{b}^{\mathrm{3}} {c}^{\mathrm{3}} +{c}^{\mathrm{3}} {a}^{\mathrm{3}} \right) \\ $$$$=\mathrm{44}^{\mathrm{2}} −\mathrm{2}\left[\left({ab}+{bc}+{ca}\right)^{\mathrm{3}} −\mathrm{3}{abc}\left({ab}+{bc}+{ca}\right)\left({a}+{b}+{c}\right)\right. \\ $$$$\left.+\mathrm{3}\left({abc}\right)^{\mathrm{2}} \right] \\ $$$$=\mathrm{44}^{\mathrm{2}} −\mathrm{2}\left[\mathrm{22}^{\mathrm{3}} −\mathrm{3}\left(\mathrm{20}\right)\left(\mathrm{22}\right)\left(\mathrm{8}\right)+\mathrm{3}\left(\mathrm{20}\right)^{\mathrm{2}} \right]=−\mathrm{640} \\ $$
Commented by Ghisom last updated on 05/Oct/24
a, b, c ∈C ⇒ x^3 =512 ⇒ x=8∨x=−4±4(√3)i ⇒ [a, b, c ordered by angle] (1) x=8∧y=22∧z=20 (s−a)(s+b)(s+c)= =s^3 −8s^2 +22s−20=0 a=2 b=3+i c=3−i a^6 +b^6 +c^6 =−640 ★ (2) x=−4−4(√3)i∧y=−26+16(√3)i∧z=140+40(√3)i (s−a)(s+b)(s+c)= =s^3 +4(1+(√3)i)s^2 −2(13−8(√3)i)s−20(7+2(√3)i)=0 a≈−5.51488+.736675i b≈−2.60820−5.29849i c≈4.12308−2.36639i a^6 +b^6 +c^6 =48320−8640(√3)i ★ (3) x=−4+4(√3)i∧y=−26−16(√3)i∧z=140−40(√3)i (s−a)(s+b)(s+c)= =s^3 +4(1−(√3)i)s^2 −2(13+8(√3)i)s−20(7−2(√3)i)=0 a≈4.12308+2.36639i b≈−2.60820+5.29849i c≈−5.51488−.736675i a^6 +b^6 +c^6 =48320+8640(√3)i ★
$${a},\:{b},\:{c}\:\in\mathbb{C} \\ $$$$\Rightarrow \\ $$$${x}^{\mathrm{3}} =\mathrm{512}\:\Rightarrow\:{x}=\mathrm{8}\vee{x}=−\mathrm{4}\pm\mathrm{4}\sqrt{\mathrm{3}}\mathrm{i} \\ $$$$\Rightarrow \\ $$$$ \\ $$$$\left[{a},\:{b},\:{c}\:\mathrm{ordered}\:\mathrm{by}\:\mathrm{angle}\right] \\ $$$$ \\ $$$$\left(\mathrm{1}\right) \\ $$$${x}=\mathrm{8}\wedge{y}=\mathrm{22}\wedge{z}=\mathrm{20} \\ $$$$\left({s}−{a}\right)\left({s}+{b}\right)\left({s}+{c}\right)= \\ $$$$={s}^{\mathrm{3}} −\mathrm{8}{s}^{\mathrm{2}} +\mathrm{22}{s}−\mathrm{20}=\mathrm{0} \\ $$$${a}=\mathrm{2} \\ $$$${b}=\mathrm{3}+\mathrm{i} \\ $$$${c}=\mathrm{3}−\mathrm{i} \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =−\mathrm{640}\:\bigstar \\ $$$$ \\ $$$$\left(\mathrm{2}\right) \\ $$$${x}=−\mathrm{4}−\mathrm{4}\sqrt{\mathrm{3}}\mathrm{i}\wedge{y}=−\mathrm{26}+\mathrm{16}\sqrt{\mathrm{3}}\mathrm{i}\wedge{z}=\mathrm{140}+\mathrm{40}\sqrt{\mathrm{3}}\mathrm{i} \\ $$$$\left({s}−{a}\right)\left({s}+{b}\right)\left({s}+{c}\right)= \\ $$$$={s}^{\mathrm{3}} +\mathrm{4}\left(\mathrm{1}+\sqrt{\mathrm{3}}\mathrm{i}\right){s}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{13}−\mathrm{8}\sqrt{\mathrm{3}}\mathrm{i}\right){s}−\mathrm{20}\left(\mathrm{7}+\mathrm{2}\sqrt{\mathrm{3}}\mathrm{i}\right)=\mathrm{0} \\ $$$${a}\approx−\mathrm{5}.\mathrm{51488}+.\mathrm{736675i} \\ $$$${b}\approx−\mathrm{2}.\mathrm{60820}−\mathrm{5}.\mathrm{29849i} \\ $$$${c}\approx\mathrm{4}.\mathrm{12308}−\mathrm{2}.\mathrm{36639i} \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =\mathrm{48320}−\mathrm{8640}\sqrt{\mathrm{3}}\mathrm{i}\:\bigstar \\ $$$$ \\ $$$$\left(\mathrm{3}\right) \\ $$$${x}=−\mathrm{4}+\mathrm{4}\sqrt{\mathrm{3}}\mathrm{i}\wedge{y}=−\mathrm{26}−\mathrm{16}\sqrt{\mathrm{3}}\mathrm{i}\wedge{z}=\mathrm{140}−\mathrm{40}\sqrt{\mathrm{3}}\mathrm{i} \\ $$$$\left({s}−{a}\right)\left({s}+{b}\right)\left({s}+{c}\right)= \\ $$$$={s}^{\mathrm{3}} +\mathrm{4}\left(\mathrm{1}−\sqrt{\mathrm{3}}\mathrm{i}\right){s}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{13}+\mathrm{8}\sqrt{\mathrm{3}}\mathrm{i}\right){s}−\mathrm{20}\left(\mathrm{7}−\mathrm{2}\sqrt{\mathrm{3}}\mathrm{i}\right)=\mathrm{0} \\ $$$${a}\approx\mathrm{4}.\mathrm{12308}+\mathrm{2}.\mathrm{36639i} \\ $$$${b}\approx−\mathrm{2}.\mathrm{60820}+\mathrm{5}.\mathrm{29849i} \\ $$$${c}\approx−\mathrm{5}.\mathrm{51488}−.\mathrm{736675i} \\ $$$${a}^{\mathrm{6}} +{b}^{\mathrm{6}} +{c}^{\mathrm{6}} =\mathrm{48320}+\mathrm{8640}\sqrt{\mathrm{3}}\mathrm{i}\:\bigstar \\ $$

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