How do you write the partial fraction decomposition of the rational expression # x / (x² - x - 2)#?

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Answer 1 · 2016-10-22T17:15:24

#x/(x^2 - x - 2) = 1/(3(x + 1)) + 2/(3(x - 2))#

Factor the denominator:

#x^2 - x - 2 = (x + 1)(x - 2)#

Write each factor as a term with an unknown coefficient:

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

Multiply both sides by #(x + 1)(x - 2)#

#x = A(x -2) + B(x + 1)#

Make the term containing B become 0 by setting #x = -1#:

#-1 = A(-1 -2) + B(-1 + 1)#

#-1 = A(-3)#

#A = 1/3#

#x = 1/3(x -2) + B(x + 1)#

Make the term containing #1/3# become zero by setting #x = 2#

#2 = 1/3(2 -2) + B(2 + 1)#

#B = 2/3#

Check:

#(1/3)(1/(x + 1)) + 2/3(1/(x - 2)) = #

#(1/3)(1/(x + 1))((x - 2)/(x - 2)) + 2/3(1/(x - 2))((x + 1)/(x + 1)) = #

#(x - 2 + 2x + 2)/(3(x - 2)(x + 1)) = #

#(3x)/(3(x - 2)(x + 1)) = #

#x/((x - 2)(x + 1)) = #

#x/(x^2 - x -2)# This checks.