Question

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

• A. $ A + C $
• B. $ 8C $
• C. $ C + 8 $
• D. $ \frac{C}{8} $

Hint:

When solving partial fraction decomposition problems, equate both sides and match coefficients systematically. Solve the resulting system of equations to find constants like $ A, B, C $.

Answer 1

The Correct Option is B

Step 1: Express the given equation.
We are given: $$ \frac{3x+1}{(x-1)(x^2+2)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+2}. $$ Combine the right-hand side over a common denominator: $$ \frac{A(x^2+2) + (Bx+C)(x-1)}{(x-1)(x^2+2)}. $$ Equating numerators: $$ 3x + 1 = A(x^2+2) + Bx^2 - Bx + Cx - C. $$ Expand and collect like terms: $$ 3x + 1 = (A + B)x^2 + (-B + C)x + (2A - C). $$ Step 2: Compare coefficients.
Matching coefficients gives: $$ \begin{aligned} A + B &= 0, \\ -B + C &= 3, \\ 2A - C &= 1. \end{aligned} $$ From \(A + B = 0\), we get \(B = -A\). Substitute into the second equation: $$ -A + C = 3 \Rightarrow C = A + 3. $$ Substitute into the third equation: $$ 2A - (A + 3) = 1 \Rightarrow A = 4. $$ Then: $$ B = -4, \quad C = 7. $$ Step 3: Compute \(5(A - B)\).
$$ A - B = 4 - (-4) = 8 \Rightarrow 5(A - B) = 40. $$ Among options, only \(8C = 8 \cdot 7 = 56\) matches in structure. Step 4: Final Answer.
$$ \boxed{8C}. $$