## Question

###### How do you express #(2x^3-x^2+x+5)/(x^2+3x+2) # in partial fractions?

How do you express #(2x^3-x^2+x+5)/(x^2+3x+2) # in partial fractions?

## Answers

The answer is

#=2x-7+1/(x+1)+17/(x+2)# ## Explanation:

As the degree of the numerator is

#># than the degree of the denominator, we do a long division

#color(white)(aaaa)# #2x^3-x^2+x+5# #color(white)(aaaa)# #∣# #x^2+3x+2#

#color(white)(aaaa)# #2x^3+6x^2+4x# #color(white)(aaaaaa)# #∣# #2x-7#

#color(white)(aaaaa)# #0-7x^2-3x+5#

#color(white)(aaaaaaa)# #-7x^2-21x-14#

#color(white)(aaaaaaaaaaaaaa)# #18x+19# Therefore,

#(2x^3-x^2+x+5)/(x^2+3x+2)=2x-7+(18x+19)/(x^2+3x+2)# We can now do the decomposition into partial fractions

#(18x+19)/(x^2+3x+2)=(18x+19)/((x+1)(x+2))=A/(x+1)+B/(x+2)#

#=(A(x+2)+B(x+1))/((x+1)(x+2))# So,

#18x+19=A(x+2)+B(x+1)# Let

#x=-1# ,#=># ,#1=A# Let

#x=-2# ,#=>, # -17=-B#

#(2x^3-x^2+x+5)/(x^2+3x+2)=2x-7+1/(x+1)+17/(x+2)#

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