Question

If \[ \frac{x+1}{(x - 1)^2(x^2 + 1)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{Cx + D}{x^2 + 1}, \] then \[ \sqrt{3A^2 + 4D^2 + 5C^2 + B^2} = ? \]

• A. \(\dfrac{3}{2}\)
• B. \(\dfrac{1}{2}\)
• C. 1
• D. 2

Hint:

In partial fraction decomposition, equating coefficients after combining the expressions simplifies solving for unknown constants. Always match coefficients of like powers of \(x\) to form a solvable system.

Answer 1

The Correct Option is D

We are given the rational function: \[ \frac{x+1}{(x - 1)^2(x^2 + 1)} \] and its decomposition: \[ \frac{x+1}{(x - 1)^2(x^2 + 1)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{Cx + D}{x^2 + 1} \] Step 1: Combine RHS into a single fraction
Take LHS: \[ \frac{x+1}{(x - 1)^2(x^2 + 1)} \] Take common denominator on RHS: \[ \frac{A(x - 1)(x^2 + 1) + B(x^2 + 1) + (Cx + D)(x - 1)^2}{(x - 1)^2(x^2 + 1)} \] Step 2: Equating numerators Numerator of RHS becomes: \[ A(x - 1)(x^2 + 1) + B(x^2 + 1) + (Cx + D)(x^2 - 2x + 1) \] Now expand all three terms: 1. \(A(x - 1)(x^2 + 1) = A(x^3 + x - x^2 - 1) = Ax^3 - Ax^2 + Ax - A\) 2. \(B(x^2 + 1) = Bx^2 + B\) 3. \((Cx + D)(x^2 - 2x + 1)\) \[ = Cx(x^2 - 2x + 1) + D(x^2 - 2x + 1) = Cx^3 - 2Cx^2 + Cx + Dx^2 - 2Dx + D \] Now collect all terms: \[ Ax^3 - Ax^2 + Ax - A + Bx^2 + B + Cx^3 - 2Cx^2 + Cx + Dx^2 - 2Dx + D \] Group like terms: - \(x^3:\ A + C\) - \(x^2:\ -A + B - 2C + D\) - \(x:\ A + C - 2D\) - Constant: \(-A + B + D\) Now compare with LHS numerator: \(x + 1\) This means: \[ A + C = 0 \tag{1} \] \[ -A + B - 2C + D = 0 \tag{2} \] \[ A + C - 2D = 1 \tag{3} \] \[ -A + B + D = 1 \tag{4} \] Step 3: Solve system of equations From (1): \(C = -A\) Substitute into (3): \[ A - A - 2D = 1 \Rightarrow -2D = 1 \Rightarrow D = -\frac{1}{2} \] Substitute into (2): \[ -A + B + 2A + D = 0 \Rightarrow A + B + D = 0 \Rightarrow A + B - \frac{1}{2} = 0 \Rightarrow B = -A + \frac{1}{2} \] Now use (4): \[ -A + B + D = 1 \Rightarrow -A + (-A + \frac{1}{2}) - \frac{1}{2} = 1 \Rightarrow -2A = 1 \Rightarrow A = -\frac{1}{2} \] Now calculate: \[ C = -A = \frac{1}{2}, D = -\frac{1}{2}, B = -A + \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1 \] Step 4: Evaluate expression \[ \sqrt{3A^2 + 4D^2 + 5C^2 + B^2} = \sqrt{3\left(\frac{1}{4}\right) + 4\left(\frac{1}{4}\right) + 5\left(\frac{1}{4}\right) + (1)^2} = \sqrt{\frac{3 + 4 + 5}{4} + 1} = \sqrt{\frac{12}{4} + 1} = \sqrt{3 + 1} = \sqrt{4} = 2 \] \[ \boxed{2} \]