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
Step 1: Rewrite the equation. The given equation is z^4 = 1. We can rewrite this as z^4 - 1 = 0. Step 2: Factor the equation using the difference of squares formula, a^2 - b^2 = (a-b)(a+b). Here, a = z^2 and b = 1. (z^2)^2 - 1^2 = 0 (z^2 - 1)(z^2 + 1) = 0 Step 3: Factor each term further. The first term, z^2 - 1, is another difference of squares: z^2 - 1 = (z-1)(z+1) The second term, z^2 + 1, can be factored using the imaginary unit i, where i^2 = -1: z^2 + 1 = z^2 - (-1) = z^2 - i^2 = (z-i)(z+i) Step 4: Combine the factors to find all roots. Substituting the factored terms back into the equation: (z-1)(z+1)(z-i)(z+i) = 0 Setting each factor to zero gives the distinct roots: z-1 = 0 z = 1 z+1 = 0 z = -1 z-i = 0 z = i z+i = 0 z = -i The four distinct roots are 1, -1, i, -i. Step 5: Compare the roots with the given options. A. (1, -1, i, 2i) B. (3, 2, 1, -2) C. (1, -1, i, -i) D. (i, -2i, -3i, -4i) The calculated roots match option C. The final answer is C. (1, -1, i, -i).