How do you write the partial fraction decomposition of the rational expression # 10/[(x-1)(x^2+9)] #?

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Answer 1 · 2018-01-27T20:15:56

#1/(x-1)+(-x-1)/(x^2+9)#

We can rewrite the function as:

#10/((x-1)(x^2+9))=A/(x-1)+(Bx+C)/(x^2+9)#

and now we need to work out #A# and #B#.

Multiply both sides by #(x-1)(x^2+9)# to get:

#10 = A(x^2+9)+(Bx+C)(x-1)#

Let #x=1# to cancel the second term and get:

#10=A(1^2+9)=10A implies A=1#

Now let #x=0# to cancel the #B#, we get:

#10 = 1*(0^2+9)+C(0-1)#

#implies 10=9-C implies C = -1#

Finally, choose any other value of #x# to get #B#, say #x=2#

#implies 10 =1*(2^2+9)+(B(2)-1)(2-1) #

#implies 10 = 13+2B-1#

#implies B = -1#

Now putting our values in:

#10/((x-1)(x^2+9))=1/(x-1)+(-x-1)/(x^2+9)#