Partial Decomposition of Fractions

In this section we covered a variety of topics, and, since I had not created a post for Partial Decomposition of Fractions, I will make that the subject of this post. Below, I have listed the steps for solving such a problem:

1. Identify the factors of the denominator:

 5x - 4    
x² - x - 2

 5x - 4      
(x-2)(x+1)

2. Create a new fraction for each of those factors:

   A  
(x-2)
+
   B  
(x+1)

3. Multiply this new set of fractions by the denominator of the first fraction:

A(x+1) + B(x-2)

4. Simplify and combine like terms:

Ax + A + Bx - 2B

5. Factor:

(A+B)x + A - 2B

6. Equate the coefficients to their counterparts in the numerator of the first equation, and solve for their values:

A + B = 5
A - 2B = 4

7. Solve as a system of equations:

A = 14/3
B = 1/3

8. Substitute these values into the equation in Step 2, and that will be the answer:

  14  
3x-6
+
   1   
3x+3