To express \(\frac{7x^2 - 3x + 2}{x(x^2 + 1)}\) in partial fractions, we assume the form:
\(\frac{A}{x} + \frac{Bx + C}{x^2 + 1}\)
Multiply through by the denominator \(x(x^2 + 1)\) to clear the fractions:
\(7x^2 - 3x + 2 = A(x^2 + 1) + (Bx + C)x\)
Expanding the right side gives:
\(7x^2 - 3x + 2 = Ax^2 + A + Bx^2 + Cx\)
Combine like terms:
\(7x^2 - 3x + 2 = (A + B)x^2 + Cx + A\)
Equate coefficients:
1. \(A + B = 7\)
2. \(C = -3\)
3. \(A = 2\)
From equation 3, \(A = 2\).
Substitute \(A = 2\) into equation 1:
\(2 + B = 7\)
\(B = 5\)
Thus, the partial fraction decomposition is:
\(\frac{2}{x} + \frac{5x - 3}{x^2 + 1}\)
