How do you write the partial fraction decomposition of the rational expression # (x^2+8)/(x^2-5x+6)#?

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Answer 1 · 2015-12-15T20:49:24

#(x^2+8)/(x^2-5x+6)=1-(5x+2)/(x^2+8)#

Since the degree of the denominator is not more than the degree of the numerator, we first long divide the expression to obtain that

#(x^2+8)/(x^2-5x+6)=1+(-5x-2)/(x^2+8)#

The denominator #x^2+8# is an irreducible quadratic factor and hence has a partial fraction form given by

#(-5x-2)/(x^2+8)=(Ax+B)/(x^2+8)#

By comparing terms in the numerator of these equivalent fractions, it is then clear that #A=-5 and B=-2#.

Hence the partial fraction decomposition for the original expression is

#(x^2+8)/(x^2-5x+6)=1+(-5x-2)/(x^2+8)#