The partial fraction decomposition of \( \frac{8 x-25}{8 x^{2}+2 x-21} \)

To decompose the fraction \( \frac{8x - 25}{8x^2 + 2x - 21} \), you first need to factor the quadratic in the denominator. Factoring gives you \( 8x^2 + 2x - 21 = (4x - 3)(2x + 7) \). Now, express the fraction as \( \frac{A}{4x - 3} + \frac{B}{2x + 7} \). Multiply through by the denominator to get \( 8x - 25 = A(2x + 7) + B(4x - 3) \). By choosing suitable values for \( x \), you can solve for \( A \) and \( B \), which will lead you to the complete partial fraction decomposition. To find the values of \( A \) and \( B \), you can use techniques like substitution or equate coefficients. Setting \( x = \frac{3}{4} \) neatly eliminates \( B \) and lets you solve for \( A \), while \( x = -\frac{7}{2} \) does the opposite. Be careful with your arithmetic! Common mistakes often include misapplying the factorization or distributing incorrectly when solving for \( A \) and \( B \).