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
Hey agyenim, good to see you again. Step 1: Factor the denominator. The denominator x^2-16 is a difference of squares, which can be factored as (x-4)(x+4). (1)/(x^2-16) = (1)/((x-4)(x+4)) Step 2: Set up the partial fraction decomposition. For distinct linear factors, the form is: (1)/((x-4)(x+4)) = (A)/(x-4) + (B)/(x+4) Step 3: Clear the denominators. Multiply both sides by (x-4)(x+4): 1 = A(x+4) + B(x-4) Step 4: Solve for A and B. To find A, set x=4: 1 = A(4+4) + B(4-4) 1 = 8A + 0 A = (1)/(8) To find B, set x=-4: 1 = A(-4+4) + B(-4-4) 1 = 0 - 8B B = -(1)/(8) Step 5: Substitute the values of A and B back into the partial fraction form. (1)/(x^2-16) = (1)/(8)x-4 + (-1)/(8)x+4 (1)/(x^2-16) = (1)/(8(x-4)) - (1)/(8(x+4)) The expression in partial fractions is: (1)/(8(x-4)) - (1)/(8(x+4)) Drop the next question!