Question 216791
Split into a sum of two rational expressions with unlike denominators:


{{{(2x+3)/(x^2+3x+2)}}}


Step 1.  This problem is called partial fraction expansion.


Factor the denominator where we need two integers m and n so that the sum is two and their product is 2.  That is m+n=3 and m*n=2.  


Step 2.  The two integers are 1 and 2.  So the denominator is factored as 


{{{x^2+3x+2=(x+1)(x+2)}}}


Step 3.   Substitute the factored denominator into the given equation:


{{{(2x+3)/(x^2+3x+2)=(2x+3)/(x+1)(x+2)}}}


Step 4.  Now what we want is the following 


{{{(2x+3)/(x+1)(x+2)=A/(x+1)+B/(x+2)}}}  where A and B are constants.


Once we find A and B we completed the problem.


Step 5.  So take the equation in Step 4 and multiply by (x+1)(x+2) to both sides of the equations in order to get rid of the denominators and to find A and B.  


{{{(x+1)(x+2)(2x+3)/(x+1)(x+2)=(x+1)(x+2)*A/(x+1)+(x+1)(x+2)*B/(x+2)}}}


Step 6.  This will simplify by canceling common factors in both the numerator and denominator 


{{{2x+3=A*(x+2)+B*(x+1)=Ax+2A+Bx+B}}}


{{{2x+3=(A+B)x+2A+B}}}


Step 7.  Now compare the left side and right side of the equation in Step 6:  Look at the x terms they must be equal and the numbers must be equal as well.  That is,


{{{2x=(A+B)x}}} or {{{2=A+B}}}  Equation A1

{{{3=2A+B}}}  Equation B1



We have two Equations A1 and B1 and two unknown variables A and B:  


I can see that A=1 and B=1 will satisfy Equations A1 and B1


But, let's so through the process in finding A and B.


Let's subtract Equation B1 from Equation A1 to get:


2-3=A-2A+B-B or -1=-A or A=1  Then using Equation A1 to solve for B=2-1=1.


So A=1 and B=1.


Step 8.  ANSWER:  {{{(2x+3)/(x^2+3x+2)=(2x+3)/(x+1)(x+2)=1/(x+1)+1/(x+2)}}}  


I hope the above steps were helpful. 


For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra or for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.


And good luck in your studies!


Respectfully,
Dr J