Find a polynomial function \( P \) of lowest degree, having rational coefficients, with the given zero. \( 2 i \)

To find a polynomial function \( P \) with rational coefficients that has \( 2i \) as a zero, we also need to consider the complex conjugate, \( -2i \), since polynomial coefficients must be rational. The simplest polynomial \( P \) can be formed by using these two roots: \[ P(x) = (x - 2i)(x + 2i) \] This simplifies to: \[ P(x) = x^2 + 4 \] So, the polynomial function \( P(x) = x^2 + 4 \) is the answer. This polynomial has a degree of 2, which is the lowest degree possible with the given root \( 2i \) and its conjugate. For extra fun, did you know the roots of this polynomial are complex? That means when you graph it, it doesn't touch the x-axis! Instead, it sits entirely above the x-axis, showcasing the beauty of imaginary numbers in mathematics.