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
Step 1: Factor the denominator. Find two numbers that multiply to 12 and add to 7: 3 and 4. n^2 + 7n + 12 = (n + 3)(n + 4) Step 2: Set up the partial fraction decomposition. (27)/((n + 3)(n + 4)) = (a)/(n + 3) + (b)/(n + 4) Step 3: Multiply both sides by (n + 3)(n + 4). 27 = a(n + 4) + b(n + 3) Step 4: Solve for a. Substitute n = -3. 27 = a(-3 + 4) + b(-3 + 3) 27 = a(1) + b(0) a = 27 Step 5: Solve for b. Substitute n = -4. 27 = a(-4 + 4) + b(-4 + 3) 27 = a(0) + b(-1) b = -27 Step 6: Write the decomposition. (27)/(n + 3) + (-27)/(n + 4) = (27)/(n + 3) - (27)/(n + 4) Final answer: (27)/(n+3) - (27)/(n+4)