How do you write the partial fraction decomposition of the rational expression #[((3x^2)+3x+12) / ((x-5)(x^2+9))]#?

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Answer 1 · 2017-01-27T10:39:52

The answer is #=3/(x-5)+3/(x^2+9)#

Let's work out the decomposition into partial fractions

#(3x^2+3x+12)/((x-5)(x^2+9))=A/(x-5)+(Bx+C)/(x^2+9)#

#=(A(x^2+9)+(Bx+C)(x-5))/((x-5)(x^2+9))#

Since the denominators are the same, we can compare the numerators

#3x^2+3x+12=A(x^2+9)+(Bx+C)(x-5)#

Let #x=5#, #=>#, #75+15+12=(25+9)A#

#=>#, #34A=102#, #=>#, #A=3#

Coefficients of #x^2#, #3=A+B#

#=>#, #B=3-A=0#

Coefficients of #x#, #=>#, #3=C#

So,

#(3x^2+3x+12)/((x-5)(x^2+9))=3/(x-5)+3/(x^2+9)#