Final answer:
The area of the square formed by the roots of the polynomial p(z) = z^4 - 1 is 2 square units. The side length of the square is √2, found by using the distance formula between the roots.
Step-by-step explanation:
To find the area of the square generated by connecting consecutive roots of the polynomial p(z) = z^4 - 1, we first need to identify the roots. This polynomial can be factored into (z^2 + 1)(z^2 - 1), which further factors into (z - i)(z + i)(z - 1)(z + 1), revealing the roots: i, -i, 1, and -1. These roots form a square on the complex plane with side length equal to the distance between two adjacent roots (for instance, the distance between 1 and i).
The distance formula for two complex numbers z1 = a + bi and z2 = c + di is √((a - c)^2 + (b - d)^2). Calculating the distance between 1 and i gives us √((1 - 0)^2 + (0 - 1)^2) = √1 + √1 = √2. Therefore, the side length of the square is √2.
To find the area of the square, we square the side length: (√2)^2 = 2. The units of this area will be in square units of whatever unit the side length was measured in, though in the context of complex numbers, units are typically not specified as we're dealing with a plane of complex numbers.