Final answer:
The partial fraction decomposition of the function 2 / (x-2)(x+1) is obtained by expressing it as A/(x-2) + B/(x+1) and solving for A and B, which result in A = 2/3 and B = -2/3. Therefore, the decomposition is 2/3/(x-2) - 2/3/(x+1).
Step-by-step explanation:
The task at hand is to find the partial fraction decomposition of the rational function 2 / (x-2)(x+1). To do this, we assume that the rational function can be broken down into a sum of fractions:
A/(x-2) + B/(x+1) = 2/(x-2)(x+1)
We will determine the values of A and B that make this equation true. To find A and B, we multiply both sides by the common denominator (x-2)(x+1), leading to:
A(x+1) + B(x-2) = 2
Now, we will substitute appropriate values for x to solve for A and B. If we put x=2, the term containing B will become zero, making it easier to solve for A:
A(2+1) = 2
A = 2/3
Similarly, by choosing x=-1, the term containing A will become zero, allowing us to solve for B:
B(-1-2) = 2
B = -2/3
Therefore, the partial fraction decomposition of 2 / (x-2)(x+1) is:
2/3/(x-2) - 2/3/(x+1)