Question

If \(\frac{2x+5}{(x-1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+3}\), then A+B =

• A. -2
• B. 2
• C. 1
• D. -1

Hint:

When asked for a sum or combination of the constants (like A+B), always try the method of comparing coefficients first. As seen in Method 2, comparing the coefficients of the highest power of the variable (in this case, \(x\)) can directly give the answer without needing to calculate the individual values of A and B. This is a very efficient exam strategy.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks us to find the sum of the constants A and B in the partial fraction decomposition of the given rational expression.

Step 2: Key Formula or Approach (Alternate Method):
Instead of solving for A and B individually using substitution, we can compare coefficients of x directly to get A+B instantly.

Step 3: Detailed Explanation:
We start with: (2x+5)/((x-1)(x+3)) = A/(x-1) + B/(x+3). Combine RHS: [A(x+3) + B(x-1)]/[(x-1)(x+3)]. Equate numerators: 2x + 5 = A(x+3) + B(x-1). Expand RHS: A(x+3) + B(x-1) = Ax + 3A + Bx - B = (A+B)x + (3A-B). Now compare coefficients of x on both sides: LHS coefficient of x is 2. RHS coefficient of x is (A+B). So, A + B = 2. We get the answer directly without needing individual values.

Step 4: Final Answer:
The value of A+B is 2.