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
You're on a roll — Step 1: Identify the polynomial and apply the Factor Theorem. The polynomial is P(x) = 2x^3 - 8x + x^2 + k. It's good practice to write it in descending powers of x: P(x) = 2x^3 + x^2 - 8x + k. The given factor is (2x+1). According to the Factor Theorem, if (2x+1) is a factor of P(x), then P(-(1)/(2)) must be equal to 0. Step 2: Substitute x = -(1)/(2) into the polynomial and set it equal to 0. P(-(1)/(2)) = 2(-(1)/(2))^3 + (-(1)/(2))^2 - 8(-(1)/(2)) + k = 0 Step 3: Simplify the terms. 2(-(1)/(8)) + ((1)/(4)) - 8(-(1)/(2)) + k = 0 -(2)/(8) + (1)/(4) + (8)/(2) + k = 0 -(1)/(4) + (1)/(4) + 4 + k = 0 Step 4: Solve for k. 0 + 4 + k = 0 4 + k = 0 k = -4 The value of the constant k is -4. Send me the next one 📸