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Question Number 50970 by peter frank last updated on 22/Dec/18

Given 3−2i and 1+i are the two of roots of the equation ax^4 +bx^3 +cx^3 +dx+e find the values of a,b,c,d and e

$${Given}\:\mathrm{3}−\mathrm{2}{i}\:{and}\:\mathrm{1}+{i} \\ $$$${are}\:{the}\:{two}\:{of}\:{roots}\:{of} \\ $$$${the}\:{equation} \\ $$$${ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{3}} +{dx}+{e} \\ $$$${find}\:{the}\:{values}\:{of} \\ $$$${a},{b},{c},{d}\:{and}\:{e} \\ $$

Answered by peter frank last updated on 23/Dec/18

let z_(1 ) =3−2i z_(2 ) =1+i z_(1 ) ^− =3+2i z_2 ^− =1−i x^2 −(3+2i+3−2i)x+(3+2i)(3−2i)=0 x^2 −6x+13=0 ......(i) the same way x^2 −(1+i+1−i)x+(1+i)(1−i)=0 x^2 −2x+2=0 multiply (i) ×(ii) (x^2 −6x+13)(x^2 −2x+2)=0 x^4 −8x^3 +27x^2 −38x+26=0.....(iii) ↓ ax^4 +bx^3 +cx^2 +dx+26 a=1 b=−8 c=27 d=−38 e=26

$${let}\:{z}_{\mathrm{1}\:} =\mathrm{3}−\mathrm{2}{i}\:\:{z}_{\mathrm{2}\:} =\mathrm{1}+{i} \\ $$$$\overset{−} {{z}}_{\mathrm{1}\:\:} =\mathrm{3}+\mathrm{2}{i}\:\:\:\:\:\overset{−} {{z}}_{\mathrm{2}} =\mathrm{1}−{i} \\ $$$${x}^{\mathrm{2}} −\left(\mathrm{3}+\mathrm{2}{i}+\mathrm{3}−\mathrm{2}{i}\right){x}+\left(\mathrm{3}+\mathrm{2}{i}\right)\left(\mathrm{3}−\mathrm{2}{i}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{13}=\mathrm{0}\:\:\:\:......\left({i}\right) \\ $$$${the}\:{same}\:{way} \\ $$$${x}^{\mathrm{2}} −\left(\mathrm{1}+{i}+\mathrm{1}−{i}\right){x}+\left(\mathrm{1}+{i}\right)\left(\mathrm{1}−{i}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}=\mathrm{0} \\ $$$${multiply}\:\left({i}\right)\:×\left({ii}\right) \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{13}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{4}} −\mathrm{8}{x}^{\mathrm{3}} +\mathrm{27}{x}^{\mathrm{2}} −\mathrm{38}{x}+\mathrm{26}=\mathrm{0}.....\left({iii}\right) \\ $$$$\downarrow \\ $$$${ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+\mathrm{26} \\ $$$${a}=\mathrm{1}\:\:{b}=−\mathrm{8}\:\:\:{c}=\mathrm{27}\:\:\:{d}=−\mathrm{38} \\ $$$${e}=\mathrm{26} \\ $$$$ \\ $$

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