Question

If \[ \frac{x^2+3}{x^4+2x^2+9} = \frac{Ax+B}{x^2+ax+b} + \frac{Cx+D}{x^2+cx+d} \] then \( aA + bB + cC + dD = \)

• A. \( 1 \)
• B. \( 0 \)
• C. \( -1 \)
• D. \( 2 \)

Hint:

For polynomial equations where direct factorization is complex, using symmetry or known polynomial identities can simplify finding coefficients in partial fraction decomposition.

Answer 1

The Correct Option is D

Step 1: Begin by decomposing the left-hand side into partial fractions. Assume \( ax+b \) and \( cx+d \) are factors of \( x^4+2x^2+9 \), an irreducible quartic polynomial. Since the polynomial cannot be factored over the reals, assume a direct equivalence of coefficients. 
Step 2: Since \( x^4+2x^2+9 \) is factored as \( (x^2+ax+b)(x^2+cx+d) \), equate the forms to derive the coefficients.
- Equate the coefficients from the expanded form to those in \( x^4+2x^2+9 \) to solve for \( a, b, c, \) and \( d \). 
Step 3: Use the identity \( x^2+3 = (Ax+B)(x^2+cx+d) + (Cx+D)(x^2+ax+b) \) and expand.
- Match coefficients for \( x^3, x^2, x, \) and constant terms to find \( A, B, C, \) and \( D \).
Step 4: Calculate \( aA + bB + cC + dD \) based on the determined values of \( a, b, c, d, A, B, C, \) and \( D \).