SOLUTION: I am trying to help my child but algebra was many years ago :-) . I have the answer but I do not know how to get there. Any help would be appreciated. Split into a sum of two

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I am trying to help my child but algebra was many years ago :-) . I have the answer but I do not know how to get there. Any help would be appreciated. Split into a sum of two       Log On


   

Question 216791: I am trying to help my child but algebra was many years ago :-) . I have the answer but I do not know how to get there. Any help would be appreciated.
Split into a sum of two rational expressions with unlike denominators:
2x+3/(x^2+3x+2)
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Split into a sum of two rational expressions with unlike denominators:

%282x%2B3%29%2F%28x%5E2%2B3x%2B2%29

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%5E2%2B3x%2B2=%28x%2B1%29%28x%2B2%29

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

%282x%2B3%29%2F%28x%5E2%2B3x%2B2%29=%282x%2B3%29%2F%28x%2B1%29%28x%2B2%29

Step 4. Now what we want is the following

%282x%2B3%29%2F%28x%2B1%29%28x%2B2%29=A%2F%28x%2B1%29%2BB%2F%28x%2B2%29 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.



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

2x%2B3=A%2A%28x%2B2%29%2BB%2A%28x%2B1%29=Ax%2B2A%2BBx%2BB

2x%2B3=%28A%2BB%29x%2B2A%2BB

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=%28A%2BB%29x or 2=A%2BB Equation A1
3=2A%2BB 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:

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