Final answer:
The equation involves third roots of unity and requires using properties of complex numbers to simplify expressions. However, after simplifying we get ω = 1/5, which is not possible because ω is a root of unity and cannot take this value. Hence, the expression has no solution under the given constraints.
Step-by-step explanation:
The question deals with third roots of unity, which are the solutions of z³ = 1 other than z = 1. If ω is a third root of unity, then ω³ = 1 and ω ≠ 1. To find the value of p for the expression (ω⁴ - 3ω² + 1) = (2 - ω + ω²¹¹), we start by recognizing that ω² + ω + 1 = 0, since ω is a root of z³ = 1. From this equation, we deduce that ω² = -ω - 1 and ω³ = 1. We can substitute these back into the expressions to find p.
Let's solve the first expression ω⁴ - 3ω² + 1. Since ω⁴ = ω³ × ω = 1 × ω = ω and ω² = -ω - 1, we can replace ω⁴ with ω and ω² with -ω - 1 in our equation, yielding ω - 3(-ω - 1) + 1 = 4ω + 2.
Subsequently, the second expression (2 - ω + ω²¹¹) simplifies using ω³ = 1, which implies ω¹¹ = (ω³)⁷ = 1⁷ = 1. So we replace ω²¹¹ with 1 and get 2 - ω + 1 = 3 - ω.
Therefore, p equals both 4ω + 2 and 3 - ω. Setting these two expressions equal gives us 4ω + 2 = 3 - ω, which after simplifying yields ω = 1/5. However, since ω must be a third root of unity and cannot equal 1/5, we have a contradiction. Thus, the equation has no solution under the given constraints, as ω is not equal to 1 and it cannot be 1/5.