1. If \( \frac{x^{2}+13 x+15}{(x-1)\left(x^{2}+2\right)}=\frac{A}{x-1}+\frac{B x+C}{x^{2}+2} \), then analyse and assess the descending order of \( A, B, C \).

To analyze and determine the descending order of the constants \( A \), \( B \), and \( C \) in the partial fraction decomposition: \[ \frac{x^{2} + 13x + 15}{(x-1)(x^{2}+2)} = \frac{A}{x-1} + \frac{Bx + C}{x^{2}+2} \] **Step 1: Multiply Both Sides by the Denominator** Multiply both sides by \((x-1)(x^{2}+2)\) to eliminate the denominators: \[ x^{2} + 13x + 15 = A(x^{2} + 2) + (Bx + C)(x - 1) \] **Step 2: Expand the Right-Hand Side** Expand and simplify the right-hand side: \[ A x^{2} + 2A + Bx(x) - Bx + Cx - C = (A + B)x^{2} + (-B + C)x + (2A - C) \] **Step 3: Set Up the System of Equations** Equate the coefficients of corresponding powers of \( x \): \[ \begin{cases} A + B = 1 & \text{(Coefficient of } x^{2}) \\ -B + C = 13 & \text{(Coefficient of } x) \\ 2A - C = 15 & \text{(Constant term)} \end{cases} \] **Step 4: Solve the System** 1. From the first equation: \( A = 1 - B \). 2. Substitute \( A = 1 - B \) into the third equation: \[ 2(1 - B) - C = 15 \\ 2 - 2B - C = 15 \\ -2B - C = 13 \] 3. From the second equation: \( C = B + 13 \). 4. Substitute \( C = B + 13 \) into \( -2B - C = 13 \): \[ -2B - (B + 13) = 13 \\ -3B - 13 = 13 \\ -3B = 26 \\ B = -\frac{26}{3} \] 5. Now, find \( A \) and \( C \): \[ A = 1 - \left(-\frac{26}{3}\right) = \frac{29}{3} \] \[ C = -\frac{26}{3} + 13 = \frac{13}{3} \] **Step 5: Arrange in Descending Order** Now, we have: \[ A = \frac{29}{3} \approx 9.67 \\ C = \frac{13}{3} \approx 4.33 \\ B = -\frac{26}{3} \approx -8.67 \] Arranging them from largest to smallest: \[ A > C > B \] **Conclusion:** The descending order of the constants is: \[ A, \quad C, \quad B \] **Answer:** After simplifying, A is largest, then C, and then B. The descending order is A , C, B.