0
(3-2x)/(x+1)(1-x)^2

For the above question when I find the partial fractions using the elimination method I got the following answer.

(3-2x)/(x+1)(1-x)^2 = 5/4(x+1) - 5/4(1-x) + 1/2(1-x)^2

But When I tried to find the partial fractions using the coefficient method I got a different answer. That answer is this:

(3-2x)/(x+1)(1-x)^2 = 5/4(x+1) + 5/4(1-x) + 1/2(1-x)^2

However, the correct answer is the answer I observed from the elimination method.

I have stated the calculations I have done when using the coefficient method. Can someone please point me out what is wrong with my calculations and why I got a different answer?

(3-2x)/(x+1)(1-x)^2 = A(x+1) + B/(1-x) + C/(1-x)^2

(3-2x) = A(1-x)^2 + B(x+1)(1-x) + C(x+1)


    = A(1-2x+x^2) + B (x-x^2+1-x) + C(x+1)

    = A -2Ax +Ax^2 -Bx^2+B+Cx+C

    = Ax^2 - Bx^2 -2Ax +Cx +A +B +C

    = x^2 (A-B) + x (-2A+c) + (A +B +C)





let x =1;


x^2: 0 = (A-B) --------------- 1


x: -2 = -2A+c  --------------- 2


C: 3 = A +B +C --------------- 3


1 + 3:
      3 = 2A +C --------------- 4


2 + 4:
      C = 1/2


C= (1/2) substitute in 2:


                    -2 = -2A + (1/2)

                     A = 5/4




A = (5/4) substitute in 1:


                    0 = (5/4) - B

                    B = 5/4





Thus ;


  (3-2x)/(x+1)(1-x)^2 = 5/4(x+1) + 5/4(1-x) + 1/2(1-x)^2
Pegasus008
  • 123
  • Welcome to MSE. Please use MathJax to format math on this site. To begin with, enclose all math expressions (including numbers) in $ signs. For example, $x_1^2$ will give you $x_1^2$. You'll get a much better response if your posts are easy to read. – saulspatz Mar 28 '21 at 16:27
  • At line 2 in your calculations simply choose x = 1 and x = -1 to find 2 coefficients. Then only equate $x^2$ for the third. – Paul Mar 28 '21 at 16:32

1 Answers1

0

There is nothing wrong in your computations. The partial fraction decomposition is$$\frac{3-2x}{(x+1)(x-1)^2}=\frac{5/4}{x+1}+\frac{5/4}{1-x}+\frac{1/2}{(1-x)^2}$$and the other answer is wrong.

José Carlos Santos
  • 427,504
  • But when I checked the answer with the partial fraction calculator I got the first answer.(The elimination method one) – Pegasus008 Mar 28 '21 at 18:34
  • Perhaps that what you got was$$\frac{3-2x}{(x+1)(x-1)^2}=\frac{5/4}{x+1}-\frac{5/4}{\color{red}{x-1}}+\frac{1/2}{(x-1)^2}.$$This is actually the same thing as my answer, but now there is a $-\frac54$ coefficient. – José Carlos Santos Mar 28 '21 at 18:36