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 find the values of L and k, we use the Factor Theorem. The Factor Theorem states that if (x-a) is a factor of a polynomial P(x), then P(a) = 0. Let the given polynomial be P(x) = 2kx^2 + Lx + 24. Step 1: Use the factor (x+2). If (x+2) is a factor, then P(-2) = 0. Substitute x = -2 into the polynomial: P(-2) = 2k(-2)^2 + L(-2) + 24 = 0 2k(4) - 2L + 24 = 0 8k - 2L + 24 = 0 Divide the entire equation by 2: 4k - L + 12 = 0 Rearrange to express L in terms of k: L = 4k + 12 (*) Step 2: Use the factor (x-1). If (x-1) is a factor, then P(1) = 0. Substitute x = 1 into the polynomial: P(1) = 2k(1)^2 + L(1) + 24 = 0 2k + L + 24 = 0 (**) Step 3: Solve the system of linear equations. Substitute equation () into equation (*): 2k + (4k + 12) + 24 = 0 6k + 36 = 0 6k = -36 k = (-36)/(6) k = -6 Step 4: Substitute the value of k back into equation (*) to find L. L = 4k + 12 L = 4(-6) + 12 L = -24 + 12 L = -12 Thus, the values are L = -12 and k = -6. Comparing this with the given options: A. L = -12, k = -6 B. L = -2, k = 1 C. L = -2, k = -1 D. L = 0, k = 1 The calculated values match option A. The final answer is A. L = -12, k = -6.