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Question Number 8683 by 314159 last updated on 21/Oct/16

Commented by prakash jain last updated on 21/Oct/16

x^4 −4cx^2 +6x^2 +x+1=(x−p)^2 (x−a)(x−b) =x^4 −(2p+a+b)x^3 +(ab+2ap+2bp+p^2 )x^2 −(2abp+ap^2 +bp^2 )x+abp^2 equating coefficients const term abp^2 =1⇒ab=(1/p^2 ) coeffient of x 2abp+ap^2 +bp^2 =−1 (a+b)p^2 =−1−2abp=−1−(2/p)=−((p+2)/p) (a+b)=−((p+2)/p^3 ) coefficient of x^2 ab+2ap+2bp+p^2 =6 ab+2p(a+b)+p^2 =6 (1/p^2 )−2(1/p)(((p+2)/p))+p^2 =6 1−2p−4+p^4 =6p^2 p^4 −6p^2 −2p−3=0 This method also given same relation in p that i got before. please recheck question.

x4−4cx2+6x2+x+1=(x−p)2(x−a)(x−b)=x4−(2p+a+b)x3+(ab+2ap+2bp+p2)x2−(2abp+ap2+bp2)x+abp2equatingcoefficientsconsttermabp2=1⇒ab=1p2coeffientofx2abp+ap2+bp2=−1(a+b)p2=−1−2abp=−1−2p=−p+2p(a+b)=−p+2p3coefficientofx2ab+2ap+2bp+p2=6ab+2p(a+b)+p2=61p2−21p(p+2p)+p2=61−2p−4+p4=6p2p4−6p2−2p−3=0Thismethodalsogivensamerelationinpthatigotbefore.pleaserecheckquestion.

Commented by prakash jain last updated on 21/Oct/16

p is a root p^4 −4cp^3 +6p^2 +p+1=0 c=((p^4 +6p^2 +p+1)/(4p^3 )) (i) x^4 −((p^4 +6p^2 +p+1)/p^3 )x^3 +6x^2 +x+1=0 multiply by p^3 x^4 p^3 −x^3 p^4 −(6p^2 +p+1)x^3 +(6p^3 +p^2 +p)x^2 − (p^2 +p)x^2 +(p^3 +p^2 )x−p^2 x+p^3 =0 x^3 p^3 (x−p)−(6p^2 +p+1)x^2 (x−p)−(p^2 +p)x(x−p) −p^2 (x−p)=0 (x−p)(x^3 p^3 −(6p^2 +p+1)x^2 −(p^2 +p)x−p^2 )=0 Since p is repeat root x−p is a factor of (x^3 p^3 −(6p^2 +p+1)x^2 −(p^2 +p)x−p^2 ). p^6 −(6p^2 +p+1)p^2 −(p^2 +p)p−p^2 =0 p^6 −6p^4 −p^3 −p^2 −p^3 −p^2 −p^2 =0 p^2 (p^4 −6p^2 −2p−3)=0 p^4 −6p^2 −2p−3=0 Now assuming answer in question c=(1/(3p))(p^2 +3)=((p^4 +6p^2 +p+1)/(4p^3 )) 4p^4 +12p^2 =3p^4 +18p^2 +3p+3 p^4 −6p^2 −3p−3=0 But the equation i got is p^4 −6p^2 −2p−3=0 Maybe somebody can check errors. I will recheck soon.

pisarootp4−4cp3+6p2+p+1=0c=p4+6p2+p+14p3(i)x4−p4+6p2+p+1p3x3+6x2+x+1=0multiplybyp3x4p3−x3p4−(6p2+p+1)x3+(6p3+p2+p)x2−(p2+p)x2+(p3+p2)x−p2x+p3=0x3p3(x−p)−(6p2+p+1)x2(x−p)−(p2+p)x(x−p)−p2(x−p)=0(x−p)(x3p3−(6p2+p+1)x2−(p2+p)x−p2)=0Sincepisrepeatrootx−pisafactorof(x3p3−(6p2+p+1)x2−(p2+p)x−p2).p6−(6p2+p+1)p2−(p2+p)p−p2=0p6−6p4−p3−p2−p3−p2−p2=0p2(p4−6p2−2p−3)=0p4−6p2−2p−3=0Nowassuminganswerinquestionc=13p(p2+3)=p4+6p2+p+14p34p4+12p2=3p4+18p2+3p+3p4−6p2−3p−3=0Buttheequationigotisp4−6p2−2p−3=0Maybesomebodycancheckerrors.Iwillrechecksoon.

Commented by prakash jain last updated on 23/Oct/16

314159, I think the question is not correct did u try to go thru the steps that i did.

314159,Ithinkthequestionisnotcorrectdidutrytogothruthestepsthatidid.

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