Final answer:
To find the partial fraction decomposition of the rational function (4x-1)/[(x-1)²(x+2)], factor the denominator and write the partial fraction decomposition using the appropriate form. Clear the fractions and equate the numerators, then simplify the equation and group like terms. Finally, solve the system of equations to find the values of A, B, and C.
Step-by-step explanation:
The partial fraction decomposition of the rational function (4x-1)/[(x-1)²(x+2)] can be found by following these steps:
- Factor the denominator: (x-1)²(x+2)
- Identify the distinct linear and irreducible quadratic factors: x-1, (x-1)², and x+2
- Write the partial fraction decomposition using the form A/(x-1) + B/(x-1)² + C/(x+2)
- Clear the fractions and equate the numerators: expr = A(x-1)(x+2) + B(x-1) + C(x-1)²
- Simplify the equation and group like terms: expr = (A + C)x² + (-2A + B - 2C)x + (2A - B + C)
- Equate the coefficients of like powers of x on both sides to form a system of equations
- Solve the system of equations to find the values of A, B, and C