Final answer:
To find the partial fraction decomposition of the rational function ⅔/(x-1)(x+4), assume the form A/(x-1) + B/(x+4) and solve for A and B. The result is 1/(x-1) - 1/(x+4).
Step-by-step explanation:
The task is to perform a partial fraction decomposition of the rational function ⅔/(x-1)(x+4). To do this, we assume that the decomposition will take the form: A/(x-1) + B/(x+4). Multiplying both sides of the equation by the denominator (x-1)(x+4), we get: 5 = A(x+4) + B(x-1). This is a system of linear equations in terms of A and B which can be solved simultaneously for A and B.
Setting x = 1, we eliminate B and get: 5 = A(1+4), which simplifies to A = 1. Setting x = -4, we eliminate A and get: 5 = B(-4-1), which simplifies to B = -1. Thus, the partial fraction decomposition of the given rational function is 1/(x-1) - 1/(x+4).