Find a polynomial function P of lowest degree, having rational coefficients, with the given zero. The polynomial function in expanded form is \( \mathrm{P}(\mathrm{x})=\square \)

To find a polynomial function with rational coefficients that has a given zero, we utilize the concept of conjugate pairs. If the zero provided is irrational or complex, the conjugate must also be a zero. For example, if the zero is \( 2 + \sqrt{3} \), the polynomial is formed as follows: 1. Create factors from the zeros: \( (x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) \). 2. Expand it to find the polynomial. This leads to: \[ P(x) = (x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) = (x - 2 - \sqrt{3})(x - 2 + \sqrt{3}) = (x - 2)^2 - (\sqrt{3})^2 = (x - 2)^2 - 3. \] Upon expanding, we have: \[ P(x) = (x^2 - 4x + 4 - 3) = x^2 - 4x + 1. \] So, in expanded form, the polynomial function is: \[ P(x) = x^2 - 4x + 1. \]