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If-x-3-x-3-0-has-the-roots-a-b-and-c-determine-the-monic-polynomial-with-the-roots-a-5-b-5-and-c-5-Q152396-

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Question Number 152663 by mr W last updated on 31/Aug/21
If x^3 -x+3=0 has the roots a, b and c. determine the monic polynomial with the roots a^5 , b^5 and c^5 . [Q152396]
Ifx3−x+3=0hastherootsa,bandc.determinethemonicpolynomialwiththerootsa5,b5andc5.[Q152396]
Answered by mr W last updated on 30/Aug/21
a+b+c=0 ab+bc+ca=−1 abc=−3 let p_k =a^k +b^k +c^k p_1 =e_1 =0 p_2 =e_1 p_1 −2e_2 =−2×(−1)=2 p_3 =e_1 p_2 −e_2 p_1 +3e_3 =3(−3)=−9 p_4 =e_1 p_3 −e_2 p_2 +e_3 p_1 =−(−1)2=2 p_5 =e_1 p_4 −e_2 p_3 +e_3 p_2 =−(−1)(−9)+(−3)2=−15 i.e. a^5 +b^5 +c^5 =−15 let p_k =(ab)^k +(bc)^k +(ca)^k p_1 =e_1 =ab+bc+ca=−1 e_2 =ab^2 c+bc^2 a+ca^2 b=abc(a+b+c)=0 e_3 =ab×bc×ca=(abc)^2 =9 p_2 =e_1 p_1 −2e_2 =(−1)(−1)−2×0=1 p_3 =e_1 p_2 −e_2 p_1 +3e_3 =(−1)1+3×9=26 p_4 =e_1 p_3 −e_2 p_2 +e_3 p_1 =(−1)26+9(−1)=−35 p_5 =e_1 p_4 −e_2 p_3 +e_3 p_2 =(−1)(−35)+9×1=44 i.e. a^5 b^5 +b^5 c^5 +c^5 a^5 =44 a^5 b^5 c^5 =(abc)^5 =(−3)^5 =−243 the equation with roots a^5 ,b^5 ,c^5 is x^3 +15x^2 +44x+243=0
a+b+c=0ab+bc+ca=−1abc=−3letpk=ak+bk+ckp1=e1=0p2=e1p1−2e2=−2×(−1)=2p3=e1p2−e2p1+3e3=3(−3)=−9p4=e1p3−e2p2+e3p1=−(−1)2=2p5=e1p4−e2p3+e3p2=−(−1)(−9)+(−3)2=−15i.e.a5+b5+c5=−15letpk=(ab)k+(bc)k+(ca)kp1=e1=ab+bc+ca=−1e2=ab2c+bc2a+ca2b=abc(a+b+c)=0e3=ab×bc×ca=(abc)2=9p2=e1p1−2e2=(−1)(−1)−2×0=1p3=e1p2−e2p1+3e3=(−1)1+3×9=26p4=e1p3−e2p2+e3p1=(−1)26+9(−1)=−35p5=e1p4−e2p3+e3p2=(−1)(−35)+9×1=44i.e.a5b5+b5c5+c5a5=44a5b5c5=(abc)5=(−3)5=−243theequationwithrootsa5,b5,c5isx3+15x2+44x+243=0
Commented by Tawa11 last updated on 30/Aug/21
Great sir
Greatsir

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