Write the partial fraction decomposition of the rational exp

Write the partial fraction decomposition of the rational expression 3x - 26/x^2 - x - 12 4/x - 6 - 2/x + 2 3/x - 3- 2/x - 4 3/x + 3 - 2/x - 4 5/x + 3 - 2/x - 4 2/x + 6 - 5/x - 2

Solution

(3x-26)_/(x²-x-12)

=(3x-26)_{/(x²-4x+3x-12)}

=(3x-26)_{/(x(x-4)+3(x-4))}

=(3x-26)_{/((x+3)(x-4))}

let (3x-26)_{/((x+3)(x-4))} =a_/(x+3) +b_/(x-4)

(3x-26)_{/((x+3)(x-4))} =(a(x-4)+ b(x+3))_{/((x+3)(x-4))}

(3x-26)_{/((x+3)(x-4))} =((a+b)x+(-4a+ 3b))_{/((x+3)(x-4))}

a+b = 3 , -4a+3b =-26

b =3-a

-4a+3(3-a) =-26

-4a+9-3a=-26

7a=35

a=5

b =3-a

b =3-5

b =-2

(3x-26)_{/((x+3)(x-4))} =5_/(x+3) -2_/(x-4)

therefore (3x-26)_/(x²-x-12) =5_/(x+3) -2_/(x-4)