Students will discover the complex conjugate root theorem by working this problem: Use properties of complex conjugates to show that if \(\alpha\) is a complex root of a quadratic equation \(ax^2 + bx + c = 0,\) with \(a, b, c\) being real coefficients, then so is \(\overline{\alpha}.\) Does this generalize to higher degree polynomials? Here's a fantastic explanation.