Complex Conjugate Theorem

Algebra 2 · Axiom Academy

LESSON Complex Conjugate Theorem For a polynomial with real coefficients, every non-real root is mirrored by its conjugate — so complex roots come in pairs. 1. What Is a Complex Conjugate? The complex conjugate of a number z = a + bi flips the sign of its imaginary part. Geometrically, that flip is a reflection across the real axis in the complex plane: same real part, opposite imaginary part. The number, with imaginary part +b Its conjugate, with imaginary part -b 2. The Complex Conjugate Theorem Here is the payoff. The single word that makes it true is real : the coefficients must be real numbers. So a real-coefficient polynomial can never have a lone non-real root — they always come two at a time, mirrored across the real axis. Watch the roots appear in pairs: x^2 - 4x + 13 has roots 2 + 3i and 2 - 3i — a conjugate pair. x - i has coefficients that are not all real. Its only root is i ; the conjugate -i is not a root. The word "real" is essential. Example: a cubic with real coefficients Three roots in total, and non-real ones must pair up: All real: x^3 - 6x^2 + 11x - 6 = (x-1)(x-2)(x-3) , roots 1, 2, 3 . One real + a conjugate pair: x^3 - 2x^2 + 4x - 8 = (x-2)(x^2+4) , roots . Impossible: two non-real roots that are not conjugates of each other. Conjugation plays perfectly well with + and , and it leaves every real number alone. Those two facts are the whole proof.

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