(3x^2+2x-1)/((x+5)(x^2+1))3x2+2x−1(x+5)(x2+1) becomes ...
(3x^2+2x-1)/((x+5)(x^2+1))=A/(x+5)+(Bx+C)/(x^2+1)3x2+2x−1(x+5)(x2+1)=Ax+5+Bx+Cx2+1
Multiply through by (x+5)(x^2+1)(x+5)(x2+1)
(3x^2+2x-1)=A(x^2+1)+(Bx+C)(x+5)(3x2+2x−1)=A(x2+1)+(Bx+C)(x+5)
First solve for AA
x=-5x=−5
(3(-5)^2+2(-5)-1)=A((-5)^2+1)+(B(-5)+C)((-5)+5)(3(−5)2+2(−5)−1)=A((−5)2+1)+(B(−5)+C)((−5)+5)
(3(25)-10-1)=A(25+1)+(-5B+C)(0)(3(25)−10−1)=A(25+1)+(−5B+C)(0)
(75-10-1)=A(26)+(-5B+C)(0)(75−10−1)=A(26)+(−5B+C)(0)
64=26A64=26A
64/26=(26A)/266426=26A26
32/13=A3213=A
Now solve for CC
Substitute in for xx and AA
x=0, A=32/13x=0,A=3213
(3(0)^2+2(0)-1)=(32/13)((0)^2+1)+(B(0)+C)((0)+5)(3(0)2+2(0)−1)=(3213)((0)2+1)+(B(0)+C)((0)+5)
(-1)=(32/13)(1)+(C)(5)(−1)=(3213)(1)+(C)(5)
-1=32/13+5C−1=3213+5C
-1-32/13=5C−1−3213=5C
(-13)/13-32/13=5C−1313−3213=5C
(-45)/13=5C−4513=5C
((-45)/13)/5=(5C)/5−45135=5C5
((-45)/13)*1/5=C(−4513)⋅15=C
((-9)/13)*1/1=C(−913)⋅11=C
(-9)/13=C−913=C
Substitute in AA and CC and let x=1x=1 as you solve for BB
(3(1)^2+2(1)-1)=(32/13)((1)^2+1)+(B(1)+((-9)/13))((1)+5)(3(1)2+2(1)−1)=(3213)((1)2+1)+(B(1)+(−913))((1)+5)
(3+2-1)=(32/13)(1+1)+(B-9/13)(1+5)(3+2−1)=(3213)(1+1)+(B−913)(1+5)
4=(32/13)(2)+(B-9/13)(6)4=(3213)(2)+(B−913)(6)
4=(64/13)+(6B-54/13)4=(6413)+(6B−5413)
4-64/13+54/13=6B4−6413+5413=6B
52/13-64/13+54/13=6B5213−6413+5413=6B
106/13-64/13=6B10613−6413=6B
42/13=6B4213=6B
(42/13)/6=(6B)/642136=6B6
(42/13)*1/6=B(4213)⋅16=B
7/13=B713=B
Substitute in AA, BB, and CC
(3x^2+2x-1)/((x+5)(x^2+1))=(32/13)/(x+5)+((7/13)x+(-9)/13)/(x^2+1)3x2+2x−1(x+5)(x2+1)=3213x+5+(713)x+−913x2+1
(3x^2+2x-1)/((x+5)(x^2+1))=32/(13(x+5))+(7x+(-9))/(13(x^2+1))3x2+2x−1(x+5)(x2+1)=3213(x+5)+7x+(−9)13(x2+1)
Partial-Fraction Decomposition
(3x^2+2x-1)/((x+5)(x^2+1))=32/(13(x+5))+(7x-9)/(13(x^2+1))3x2+2x−1(x+5)(x2+1)=3213(x+5)+7x−913(x2+1)
