How do you write the partial fraction decomposition of the rational expression # (x+10)/(x^2+2x-8)#?

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Answer 1 · 2015-12-09T07:35:48

The equivalent partial fraction is :
#(-1)/(x+4) + 2/(x-2)#

Given #(x+10)/(x^2 +2x-8)#

Step 1: Factor the denominator

#(x+10)/((x+4)(x-2)#

Step 2: Set up the partial faction as follows:

#(x+10)/((x+4)(x-2)) = A/(x+4) + B/(x-2) " " " " " (1)#

Step 3: Multiply both sides by the LCD, #(x+4)(x-2)#:

#(x+10) = A(x-2) +B(x+4)#
#x+ 10 = Ax - 2A + Bx+ 4B#

Step 4: Set up a system like this
#1x: " " " " A+ B= 1 " " " "(2) #
#10: " " " -2A+4B= 10 " " " "(3) #

Step 5. You can solve the system by the elimination method:

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

#+ -2A + 4B= 10 #
# 6B = 12 => B= 2#

Solve for #A# by substituting #B = 3# into #(2)#:

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

Step 6. Substitute #A# and #B# back into #(1)#:

#(x+10)/((x+4)(x-2))= (-1)/(x+4) + 2/(x-2)#