3Write the partial fraction decompositions of the given rati

3.Write the partial fraction decompositions of the given rational expressions

a

What is the partial fraction decomposition? X+ 1 Do not simplify the partial fractions. Use integers or fractions for any numbers in the expression.)

Solution

a) (x+1)/x^2(x^2 +4) = A/x +B/x^2 + (Cx+D)/(x^2 +1) + (Ex +F)/(x^2+4)^2

Multiply both sides by (x^2 +4)(x^2)

x+1 = (A+C)x^5 + (B +D)x^4 + (8A +4C+E)x^3 + (8B +4D +F)x^2 + 16Ax + 16B

1 = 16B ; 1 = 16A ; 0 = 8B +4D + F

0 = 8A + 4C +E

0 = B +D ; 0 = A +C

we get on solving : A =1/16 ; B =1/16 ; C =-1/16 ;D = -1/16 ;E =-1/4; F = -1/4

(x+1)/x^2(x^2 +4) = 1/16x + 1/16x^2 + (-x-1)/16(x^2 +1) + (-x -1)/4(x^2+4)^2

b) (x^2 -x-7)/(x^2 +1)^2 = (Ax+B)/(x^2 +1) + (Cx+D)/(x^2 +1)^2

multiply both sides by (x^2 +1)^2 and equating the coefficient:

(x^2 -x-7) = A(x^3 +x) + (x^2 +1)B + Cx +D

equations: -7 = B +D

-1 = A+C

B =1

D=0

On solving we get : A =0; B=1 ;C=-1 D = -8

(x^2-x-7)/(x^2 +1)^2 = (-x-8)/(x^2 +1)^2 +1/(x^2 +1)