To find the partial fraction decomposition of (x + 4) / (x(x + 2)), follow these steps:
1. Write the given expression in the form of partial fractions: (x + 4) / (x(x + 2)) = A/x + B/(x + 2).
2. Multiply both sides by x(x + 2) to get rid of the denominators: x + 4 = A(x + 2) + Bx.
3. Expand the right side: x + 4 = Ax + 2A + Bx.
4. Combine like terms: x + 4 = (A + B)x + 2A.
5. Equate the coefficients of like terms on both sides:
- Coefficient of x on the left side = Coefficient of x on the right side: 1 = A + B
- Constant term on the left side = Constant term on the right side: 4 = 2A
6. Solve the system of equations formed in step 5 to find the values of A and B:
- From 1 = A + B, when A = 1, then B = 0.
- From 4 = 2A, A = 2.
7. Substitute the values of A and B back into the partial fraction decomposition:
(x + 4) / (x(x + 2)) = 2/x + 0/(x + 2) = 2/x.
Therefore, the partial fraction decomposition of (x + 4) / (x(x + 2)) is 2/x.
