Final answer:
To decompose the function 2/(x-1)(x+1), we find constants A and B such that the function equals A/(x-1) + B/(x+1). By solving for A and B, we find that the partial fraction decomposition is 1/(x-1) - 1/(x+1).
Step-by-step explanation:
To find the partial fraction decomposition of the rational function 2/(x-1)(x+1), we assume that it can be represented as the sum of two simpler fractions:
A/(x-1) + B/(x+1)
Now we need to determine the constants A and B. We do this by multiplying both sides by the common denominator (x-1)(x+1) to get:
2 = A(x+1) + B(x-1)
Next, we solve for A and B by choosing suitable values for x. For instance, if we put x=1, we get:
2 = A(2), which means A = 1.
If we put x=-1, we get:
2 = -B(2), which means B = -1.
Hence, the partial fraction decomposition is:
1/(x-1) - 1/(x+1)