Question

If \[ \frac{2x^4 - 3x^2 + 4}{(x^2 + 1)(x^2 + 2)} = a + \frac{px + q}{x^2 + 1} + \frac{mx + n}{x^2 + 2}, \] then $\frac{n}{q} =$

• A. $p + m - a$
• B. $\frac{p + m}{a}$
• C. $\frac{a}{p + m}$
• D. $p + m + a$

Hint:

Partial fraction decomposition requires expanding and comparing coefficients term-by-term.

Answer 1

The Correct Option is A

Multiply both sides by $(x^2+1)(x^2+2)$:
\[ 2x^4 - 3x^2 + 4 = a(x^2+1)(x^2+2) + (px+q)(x^2+2) + (mx+n)(x^2+1). \] Expand right side and group coefficients by powers of $x$. Equate coefficients of powers $x^4, x^3, x^2, x, 1$ on both sides.
From the system of equations, solve for $a, p, q, m, n$. Finally, compute ratio $\frac{n}{q} = p + m - a$.