How do you write the partial fraction decomposition of the rational expression #5 / (x^2 + x - 6)#?

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Answer 1 · 2015-12-21T09:50:58

Factorize and solve

First step is to factorize the denominator.

#Den(5/(x^2+x-6))= x^2+x-6#
# = x^2+3x-2x-6#
# = x(x+3)-2(x+3)#
# = (x+3)(x-2)#

Now the partial fractions are
#5/(x^2+x-6) = A/(x+3)+B/(x-2)#

Solve for A & B.

Multiply both sides by denominator
#A(x-2)+B(x+3)=5#

If #x=2# we will get B

#5B=5#
#B=1#

If #x=-3# we will get A
#-5A =5#
#A = -1#

PFs are
#5/(x^2+x-6) = 1/(x-2)-1/(x+3)#