Final answer:
To decompose the fraction 15/(x(x-5)) into partial fractions, factor the denominator and equate the numerators to find A and B. The partial fraction decomposition is -3/x + 3/(x-5).
Step-by-step explanation:
To decompose the fraction 15/(x(x-5)) into partial fractions, start by factoring the denominator x(x-5) as the product of its linear factors: x(x-5). The partial fraction decomposition can be written as:
15/(x(x-5)) = A/x + B/(x-5)
To find the values of A and B, multiply the entire equation by the common denominator x(x-5) and equate the numerators:
15 = A(x-5) + Bx
Expand and combine like terms:
15 = (A + B)x - 5A
Equating the coefficients of x and the constants:
A + B = 0
-5A = 15
Solving this system of equations, we find A = -3 and B = 3. Therefore, the partial fraction decomposition of 15/(x(x-5)) is -3/x + 3/(x-5).