How do you express #1/ (x^2 + 3x -4)# in partial fractions?
# 1/5(x-1) - 1/5(x+4) #
first step is to factor the denominator :
# x^2 + 3x - 4 ) = (x+4)(x-1) #
since these factors are linear the the numerators of the partial fractions will be constants , say A and B.
# rArr 1/((x+4)(x-1)) = A/(x+4 )+ B/(x-1) #
now multiply both sides by )x+4)(x-1)
so 1 = A(x-1) + B(x+4).......................................(1)
The aim now is to find the values of A and B. Note that if x = 1 , the term with A will be zero and if x = -4 the term with B will be zero.
This is the starting point for finding A and B.
let x = 1 in (1) : 1 = 5B
# rArr B = 1/5 #
let x = -4 in (1) : 1 = -5A
# rArr A = -1/5 #
# rArr 1/(x^2+3x-4) = 1/5(x-1) - 1/5(x+4) #