Question

If \[ \frac{x^4 + 24x^2 + 28}{(x^2 + 1)^3} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2} + \frac{Ex + F}{(x^2 + 1)^3} \] then the value of $A + B + C + D + E + F =$ ?

• A. 21
• B. 22
• C. 28
• D. 29

Hint:

In partial fractions with repeated irreducible quadratics, use general terms and equate coefficients.

Answer 1

The Correct Option is C

This is a standard partial fractions decomposition. Since LHS is rational with $(x^2 + 1)^3$ in denominator, the RHS will be: \[ \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2} + \frac{Ex + F}{(x^2 + 1)^3} \] Multiply both sides by $(x^2 + 1)^3$ and equate coefficients: Resulting polynomial on RHS will have same degree as LHS, which is degree 4. Equating: \[ x^4 + 24x^2 + 28 = \text{expanded RHS} \] Compare coefficients and solve for $A, B, C, D, E, F$, their sum turns out to be $\boxed{28}$.