Question #c311f

1 Answer
May 5, 2016

Answer:

#A/B = 5/3#

We can find exact values of #A# and #B#:
#A=3/19#, #B=5/19#

Explanation:

Given: #1/[(3-5x)(2+3x)] = A/(3-5x) + B/(2+3x)#

Bring the right side to common denominator:
#A/(3-5x) + B/(2+3x) = [A(2+3x)+B(3-5x)]/[(3-5x)(2+3x)]#

Compare this with a given equality.
Obviously,
#1 = A(2+3x)+B(3-5x)# for any #x#.
Equivalently,
#(3A-5B)x+2A+3B-1=0#

Since this equality should hold for all #x#, the coefficient at #x# must be equal to zero, that is, #3A-5B=0# and the free member should also be equal to zero, that is, #2A+3B-1=0#.

From the first equality, we conclude that #A/B = 5/3#.
Using this ratio and the second equality, we can find #A# and #B#:
#2*5/3B+3B-1=0#
#10B+9B-3=0#
#B=3/19#
#A=5/19#

CHECK:
#5/19(2+3x)+3/19(3-5x) = (10+15x+9-15x)/19 = 1#
as it should.