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Decompose-the-function-P-x-x-4-2x-3-6x-2-20x-6-x-3-x-2-x-in-partial-fractions-

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Question Number 137252 by mey3nipaba last updated on 31/Mar/21
Decompose the function P(x) = ((x^4 +2x^3 +6x^2 +20x+6)/(x^3 +x^2 +x)) in partial fractions.
DecomposethefunctionP(x)=x4+2x3+6x2+20x+6x3+x2+xinpartialfractions.
Answered by bemath last updated on 31/Mar/21
⇔ x+1 + ((4x^2 +19x+6)/(x(x^2 +x+1))) = ⇔x+1 + (A/x) + ((Bx+C)/(x^2 +x+1)) ⇔ 4x^2 +19x+6 = A(x^2 +x+1)+(Bx+C)x ⇔ A = [((4x^2 +19x+6)/(x^2 +x+1)) ]_(x=0) = 6 put x=1⇒29 = 18+B+C ⇒ B+C = 11 put x=−1⇒−9= 6 −(−B+C) ⇒ −15 = B−C we get { ((B=−2)),((C=13)) :} ⇔ x+1 +(6/x)+ ((13−2x)/(x^2 +x+1))
⇔x+1+4x2+19x+6x(x2+x+1)=⇔x+1+Ax+Bx+Cx2+x+1⇔4x2+19x+6=A(x2+x+1)+(Bx+C)x⇔A=[4x2+19x+6x2+x+1]x=0=6putx=1⇒29=18+B+C⇒B+C=11putx=−1⇒−9=6−(−B+C)⇒−15=B−Cweget{B=−2C=13⇔x+1+6x+13−2xx2+x+1

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