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.
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Welcome back BIG — missed you this week. Here are the solutions to your questions. Question 1: a) Given that (x-1) is a factor of the polynomial f(x) = ax^4 - x^3 - 12x^2 - x + 2. Find the value of the constant a and verify that f(-1)=0. Step 1: Use the Factor Theorem. Since (x-1) is a factor of f(x), we know that 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 - 12 = 0 a = 12 Step 2: Verify f(-1)=0 with a=12. Substitute a=12 into f(x), so f(x) = 12x^4 - x^3 - 12x^2 - x + 2. Now, substitute x=-1 into f(x): f(-1) = 12(-1)^4 - (-1)^3 - 12(-1)^2 - (-1) + 2 f(-1) = 12(1) - (-1) - 12(1) - (-1) + 2 f(-1) = 12 + 1 - 12 + 1 + 2 f(-1) = 4 There seems to be a discrepancy with the verification part of the question, as f(-1) ≠ 0. Assuming the question meant to ask for f(1)=0 or there's a typo in the polynomial or the verification value. Based on the given polynomial and factor, f(-1)=4. b) Find the value of the constant k for which the equation x^2 + (k+1)x + k = 0 has one root, double the other. Step 1: Set up relationships between roots and coefficients. Let the roots be and 2. From Vieta's formulas: Sum of roots: + 2 = -(k+1) 3 = -(k+1) (1) Product of roots: · 2 = k 2^2 = k (2) Step 2: Solve for k. From equation (1), = -(k+1)/(3). Substitute this 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 - 5k + 2 = 0 Factor the quadratic equation: 2k^2 - 4k - k + 2 = 0 2k(k-2) - 1(k-2) = 0 (2k-1)(k-2) = 0 So, 2k-1=0 or k-2=0. k = (1)/(2) or k = 2 The values of k are k = (1)/(2) or k = 2. Question 2: i) Show that ( + 2)/(1 + + 2) = . Step 1: Apply double angle identities. Recall the identities: 2 = 2 and 2 = 2^2 - 1. Substitute these into the left-hand side (LHS) of the equation: LHS = ( + 2)/(1 + + (2^2 - 1)) Step 2: Simplify the expression. LHS = ((1 + 2))/( + 2^2) Factor out from the denominator: LHS = ((1 + 2))/((1 + 2)) Cancel the common term (1 + 2), assuming 1 + 2 ≠ 0: LHS = ()/() LHS = This matches the right-hand side (RHS). Therefore, ( + 2)/(1 + + 2) = . ii) Find the general solution of the equation 4x + 2x = 0. Step 1: Use the double angle identity for 4x. Recall 4x = 2 2x 2x. Substitute this into the equation: 2 2x 2x + 2x = 0 Step 2: Factor the equation. Factor out 2x: 2x (2 2x + 1) = 0 This gives two separate equations: 1) 2x = 0 2) 2 2x + 1 = 0 2x = -(1)/(2) Step 3: Solve for 2x in each case. For 2x = 0: The general solution for A = 0 is A = ()/(2) + n, where n Z. So, 2x = ()/(2) + n x = ()/(4) + (n)/(2) For 2x = -(1)/(2): The principal values for A = -(1)/(2) are A = -()/(6) (or (7)/(6)) and A = (11)/(6). The general solution for $