1Write the partial fraction decompositions of the given rati

1.Write the partial fraction decompositions of the given rational expressions:

a

b

c

What is the partial fraction decomposition? x-3 (x+2)x+5)2

Solution

a) (x-3)/(x+2)(x+5)^2 = A/(x+2) +B/(x+5) +C/(x+5)^2

(x-3) = A(x+5)^2 +B(x+5)(x+2) +C(x+2)

Equating the coeffiecients on both sides we get:

-3 = 25A +10B +2C

1 = 10A +7B +C

0 = A +B----> A =-5/9 ; B =5/9 ; C =8/3

On solving we get : -5/9(x+2) + 5/9(x+5) + 8/3(x+5)^2

b) x/(5x -6)(6x+1) = A/(5x-6) +B/(6x+1)

x = A(6x+1) +B(5x -6)

On equating the coefficients we get : 0 = A -6B;

1 = 6A +5B

A = 6/41 ; B =1/41

x/(5x -6)(6x+1) = 6/41(5x-6) + 1/41(6x+1)

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

x = (x-7)A + B(x-1)

equations: 0 = -7A - B

1 =A +B

on solving we get A =7/6 ; B =-1/6

x/(x^2 -8x +7) =7/6(x-7) -1/6(x-1)