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
1. (i) Step 1: Use the Factor Theorem. Given that (x-1) is a factor of f(x) = ax^4 + x^3 - 12x^2 - x + 2, by the Factor Theorem, f(1) = 0. Substitute x=1 into f(x): f(1) = a(1)^4 + (1)^3 - 12(1)^2 - (1) + 2 = 0 a + 1 - 12 - 1 + 2 = 0 a - 10 = 0 a = 10 The value of the constant a is 10. Step 2: Verify that f(-1) = 0. Now, substitute a=10 into f(x), so f(x) = 10x^4 + x^3 - 12x^2 - x + 2. Substitute x=-1 into f(x): f(-1) = 10(-1)^4 + (-1)^3 - 12(-1)^2 - (-1) + 2 f(-1) = 10(1) + (-1) - 12(1) + 1 + 2 f(-1) = 10 - 1 - 12 + 1 + 2 f(-1) = 9 - 12 + 1 + 2 f(-1) = -3 + 1 + 2 f(-1) = 0 This verifies that f(-1) = 0. 1. (ii) Step 1: Define the roots. Let the roots of the equation x^2 + (k+1)x + k = 0 be and 2. Step 2: Apply Vieta's formulas. For a quadratic equation Ax^2 + Bx + C = 0, the sum of roots is -(B)/(A) and the product of roots is (C)/(A). In this equation, A=1, B=(k+1), C=k. Sum of roots: + 2 = -(k+1) 3 = -(k+1) (1) Product of roots: · 2 = k 2^2 = k (2) Step 3: Solve the system of equations for k. From equation (1), express in terms of k: = -(k+1)/(3) Substitute this expression for into equation (2): 2(-(k+1)/(3))^2 = k 2((k+1)^2)/(9) = k 2(k^2 + 2k + 1) = 9k 2k^2 + 4k + 2 = 9k 2k^2 + 4k - 9k + 2 = 0 2k^2 - 5k + 2 = 0 Step 4: Solve the quadratic equation for k. Factor the quadratic equation: 2k^2 - 4k - k + 2 = 0 2k(k - 2) - 1(k - 2) = 0 (2k - 1)(k - 2) = 0 This gives two possible values for k: 2k - 1 = 0 k = (1)/(2) k - 2 = 0 k = 2 The values of the constant k are (1)/(2) and 2.