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Question Number 33899 by mondodotto@gmail.com last updated on 27/Apr/18

express ((4t^2 −28)/(t^4 +t^2 −6)) as a partial fraction.

express4t2−28t4+t2−6asapartialfraction.

Answered by MJS last updated on 27/Apr/18

t^4 +t^2 −6=0 t^2 =−(1/2)±(√((1/4)+6))=−(1/2)±(5/2)= −3 ∨ 2 t^4 +t^2 −6=(t^2 +3)(t^2 −2) ((4t^2 −28)/(t^4 +t^2 −6))=(A/(t^2 +3))+(B/(t^2 −2))=((A(t^2 −2)+B(t^2 +3))/(t^4 +t^2 −6)) (A+B)t^2 +(−2A+3B)=4t^2 −28 A+B=4 ∧ −2A+3B=−28 A=4−B −2(4−B)+3B=−28 B=−4 A=8 ((4t^2 −28)/(t^4 +t^2 −6))=(8/(t^2 +3))−(4/(t^2 −2))

t4+t2−6=0t2=−12±14+6=−12±52=−3∨2t4+t2−6=(t2+3)(t2−2)4t2−28t4+t2−6=At2+3+Bt2−2=A(t2−2)+B(t2+3)t4+t2−6(A+B)t2+(−2A+3B)=4t2−28A+B=4∧−2A+3B=−28A=4−B−2(4−B)+3B=−28B=−4A=84t2−28t4+t2−6=8t2+3−4t2−2

Commented by math1967 last updated on 27/Apr/18

I can also write A(t^2 −2)+B(t^2 +3)=4t^2 −28 putting t^2 =2 A×0+B×5=4×2−28 ∴B=−4 similarly we get A=8

IcanalsowriteA(t2−2)+B(t2+3)=4t2−28puttingt2=2A×0+B×5=4×2−28∴B=−4similarlywegetA=8

Answered by math1967 last updated on 27/Apr/18

((8(t^2 −2)−4(t^2 +3))/((t^2 −2)(t^2 +3))) (8/(t^2 +3)) −(4/(t^2 −2))

8(t2−2)−4(t2+3)(t2−2)(t2+3)8t2+3−4t2−2

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