Complex Zeros

Pre-Calculus · Axiom Academy

Why the complex roots of a real polynomial always arrive two at a time — a mirror-image pair across the real axis. 1. A Complex Number Is a Point A complex number z = a + bi has a real part a and an imaginary part b , where . The honest way to picture it is as a point on the complex plane : go a units along the real axis, then b units up the imaginary axis. The pair (a, b) is the number. real part a , imaginary part b 2. The Conjugate Is a Mirror Image The conjugate of z = a + bi is : keep the real part, flip the sign of the imaginary part. Geometrically that's a reflection across the real axis — same horizontal position, opposite height. The one fact that makes conjugates matter for polynomials: a number times its conjugate is always real . The imaginary parts cancel. 3. Complex Zeros Come in Pairs Here's the payoff. If f(x) is a polynomial with real coefficients and a + bi is a zero, then its conjugate a - bi is also a zero. Complex zeros of real polynomials never come alone — they arrive as mirror-image pairs across the real axis. Why it's forced: taking the conjugate of an entire expression conjugates each piece. Since the coefficients are real, conjugating them changes nothing — so . If f(z) = 0 , then too. One zero drags its mirror along. This needs real coefficients. A polynomial with complex coefficients can have a lone complex zero with no conjugate partner.

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