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decompose-the-folowing-fraction-at-R-x-1-F-x-x-3-1-x-6-2-G-x-x-2-1-x-3-x-2-x-1-2-

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Question Number 67672 by Abdo msup. last updated on 30/Aug/19
decompose the folowing fraction at R(x) 1)F(x)=(x^3 /(1−x^6 )) 2) G(x) =((x^2 +1)/(x^3 (x^2 +x+1)^2 ))
decomposethefolowingfractionatR(x)1)F(x)=x31−x62)G(x)=x2+1x3(x2+x+1)2
Commented by Abdo msup. last updated on 30/Aug/19
1)F(x) =−(x^3 /((x^3 )^2 −1)) =−(x^3 /((x^3 −1)(x^3 +1))) =((−x^3 )/((x−1)(x^2 +x+1)(x+1)(x^2 −x+1))) =(a/(x−1)) +(b/(x+1)) +((cx +d)/(x^2 +x+1)) +((ex +f)/(x^2 −x +1)) a =lim_(x→1) (x−1)F(x) =−(1/(3.2.1)) =−(1/6) b =lim_(x→−1) (x+1)F(x)=(1/((−2).3))=−(1/6) ⇒ F(x)=((−1)/(6(x−1)))−(1/(6(x+1))) +((cx+d)/(x^2 +x+1)) +((ex +f)/(x^2 −x+1)) F(−x) =−F(x) ⇒ (1/(6(x+1))) +(1/(6(x−1))) +((−cx+d)/(x^2 −x +1)) +((−ex +f)/(x^2 +x+1)) =(1/(6(x−1))) +(1/(6(x+1))) +((−cx −d)/(x^2 +x+1)) +((−ex−f)/(x^2 −x +1)) ⇒ ⇒ f=−d...
1)F(x)=−x3(x3)2−1=−x3(x3−1)(x3+1)=−x3(x−1)(x2+x+1)(x+1)(x2−x+1)=ax−1+bx+1+cx+dx2+x+1+ex+fx2−x+1a=limx→1(x−1)F(x)=−13.2.1=−16b=limx→−1(x+1)F(x)=1(−2).3=−16⇒F(x)=−16(x−1)−16(x+1)+cx+dx2+x+1+ex+fx2−x+1F(−x)=−F(x)⇒16(x+1)+16(x−1)+−cx+dx2−x+1+−ex+fx2+x+1=16(x−1)+16(x+1)+−cx−dx2+x+1+−ex−fx2−x+1⇒⇒f=−d…
Commented by Abdo msup. last updated on 30/Aug/19
⇒ F(x) =−(1/(6(x−1)))−(1/(6(x+1))) +((cx +d)/(x^2 +x+1)) +((ex−d)/(x^2 −x+1)) lim_(x→+∞) xF(x) =0 =−(1/3) +c+e ⇒e =(1/3)−c ⇒ F(x) =−(1/(6(x−1)))−(1/(6(x+1))) +((cx+d)/(x^2 +x+1)) +((((1/3)−c)x−d)/(x^2 −x +1)) to detrmine c and d we?can calculate F(2) and F(−2)...be?vontinued...
⇒F(x)=−16(x−1)−16(x+1)+cx+dx2+x+1+ex−dx2−x+1limx→+∞xF(x)=0=−13+c+e⇒e=13−c⇒F(x)=−16(x−1)−16(x+1)+cx+dx2+x+1+(13−c)x−dx2−x+1todetrminecanddwe?cancalculateF(2)andF(−2)…be?vontinued…

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