Reflection in Complex Plane?

📌 CONCEPT: The reflection of a complex number z about the origin in the complex plane is given by -z, indicating that the original complex number z is mapped to its mirror image across the origin to obtain its reflection.

📐 RULE / FORMULA: If z is a complex number and |z| = 2, then for a polynomial x^2 + bx + c with z as a root, the other root can be found using the fact that the sum of the roots of a quadratic equation ax^2 + bx + c = 0 is -b/a.

💡 WORKED EXAMPLE: Consider the polynomial x^2 + 2z + c with z as a root. By the sum of roots formula, the other root is -2z - c. If we let c = 0, then the other root becomes -2z, which is the reflection of z about the origin.

⚠️ COMMON MISTAKE: Students might mistakenly consider the reflection of z to be 2z, which is incorrect, or assume that the polynomial x^2 + bz + c has real coefficients, but this is not a requirement for the problem.