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:
To find the values of A and B, we first combine the terms on the right-hand side and then equate the numerators.
\[ \frac{A}{x-1} + \frac{B}{x+3} = \frac{A(x+3) + B(x-1)}{(x-1)(x+3)} \]
So, we have the identity:
\[ 2x+5 = A(x+3) + B(x-1) \]
We can find A and B by substituting strategic values for \(x\) (the "cover-up" method) or by comparing coefficients.

Step 3: Detailed Explanation:
We start with the identity:
\[ 2x+5 = A(x+3) + B(x-1) \]

Method 1: Cover-up Method
To find A, we substitute \(x=1\) to make the B term zero:
\[ 2(1) + 5 = A(1+3) + B(1-1) \]
\[ 7 = A(4) + B(0) \]
\[ 7 = 4A \implies A = \frac{7}{4} \]
To find B, we substitute \(x=-3\) to make the A term zero:
\[ 2(-3) + 5 = A(-3+3) + B(-3-1) \]
\[ -6 + 5 = A(0) + B(-4) \]
\[ -1 = -4B \implies B = \frac{1}{4} \]
Now, we calculate the sum A+B:
\[ A+B = \frac{7}{4} + \frac{1}{4} = \frac{8}{4} = 2 \]

Method 2: Comparing Coefficients
Expand the right side of the identity:
\[ 2x+5 = Ax + 3A + Bx - B \]
\[ 2x+5 = (A+B)x + (3A-B) \]
Now, compare the coefficients of \(x\) and the constant terms on both sides.
Comparing coefficients of \(x\):
\[ A+B = 2 \]
Comparing constant terms:
\[ 3A-B = 5 \]
From the first equation, we directly get the required value \(A+B=2\). We don't even need to solve for A and B individually.

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