This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To expand the quotient into partial fractions, we set up the decomposition as follows: Step 1: Set up the partial fraction decomposition. The denominator has two distinct linear factors, (x-3) and (x-2). (5x - 13)/((x - 3)(x - 2)) = (A)/(x - 3) + (B)/(x - 2) Step 2: Clear the denominators. Multiply both sides by (x - 3)(x - 2): 5x - 13 = A(x - 2) + B(x - 3) Step 3: Solve for A and B. To find A, let x = 3: 5(3) - 13 = A(3 - 2) + B(3 - 3) 15 - 13 = A(1) + B(0) 2 = A To find B, let x = 2: 5(2) - 13 = A(2 - 2) + B(2 - 3) 10 - 13 = A(0) + B(-1) -3 = -B B = 3 Step 4: Write the final partial fraction decomposition. Substitute the values of A and B back into the decomposition: (5x - 13)/((x - 3)(x - 2)) = (2)/(x - 3) + (3)/(x - 2) The expanded quotient is: (2)/(x - 3) + (3)/(x - 2)