Every Polynomial Has Roots

Algebra 2 · Axiom Academy

A polynomial of degree has exactly n roots in the complex numbers — counted with multiplicity. Some you can see on the graph; some hide in the complex plane. A real root is where a graph meets the x -axis. Slide the parabola y = x^2 + k upward: its two real crossings drift together, merge into a single double root , then leave the axis entirely — yet it is always degree 2 . So a degree- 2 polynomial can show 2 , 1 , or 0 real crossings. The count of real crossings changes — but the degree never does. So where do the missing roots go? The polynomial x^2 + 1 never touches the real x -axis — on the left the curve stays a full unit above it, so it has 0 real roots. But the roots are not gone. On the right, in the complex plane , they sit at i and -i : a conjugate pair , mirror images across the real axis. A complex root has no place on the real graph — it needs its own plane. The pair are the degree- 2 polynomial's two roots. Every polynomial of degree factors into exactly n linear factors over — so it has exactly n roots, counted with multiplicity. Step the degree up and count: the number of factors and the number of roots always match the degree. Real roots land on the real axis; non-real roots arrive in conjugate pairs. Multiplicity counts repeats (a double root is two); conjugate pairs cover the non-real ones. Together they always total the degree. The Fundamental Theorem of Algebra

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