Complex Roots and Polynomials?

📌 CONCEPT: A polynomial with real coefficients that has the same roots as a given quadratic equation can be found by taking the conjugate of the complex roots and forming the product of the binomials.

📐 RULE / FORMULA: If a quadratic equation has complex roots $a + bi$ and $a - bi$, then the polynomial with real coefficients having the same roots is the product of the binomials $(x - (a + bi))$ and $(x - (a - bi))$, which simplifies to $x^2 - 2ax + (a^2 + b^2)$.

💡 WORKED EXAMPLE: Consider the quadratic equation $x^2 + 4x + 5 = 0$ with complex roots $-2 + i$ and $-2 - i$. The polynomial with real coefficients having the same roots is $(x - (-2 + i))(x - (-2 - i)) = x^2 + 4x + 5$.

⚠️ COMMON MISTAKE: Students often forget to take the conjugate of the complex roots, resulting in a polynomial with complex coefficients.