Fundamental Theorem of Algebra

Pre-Calculus · Axiom Academy

LESSON Fundamental Theorem of Algebra Every degree- n polynomial has exactly n roots in — some real, the rest complex, all counted with multiplicity. Pick any non-constant polynomial with complex coefficients. The theorem promises it has at least one complex root — and, applied repeatedly, that a degree- n polynomial has exactly n roots in (counting multiplicity). Read the degree, and you know the head-count. Apply it repeatedly a degree- n polynomial has exactly n complex roots. Here is the headline corollary. Whatever the polynomial, the degree is the root count . A quadratic has 2, a cubic has 3, a quartic has 4 — every time, once you include multiplicity and complex roots. "Counting multiplicity" means a root is counted as many times as its factor appears. The graph of f(x)=(x-2)^2 only meets the x -axis at one place, x=2 — but the factor (x-2) shows up twice , so it counts as two roots. Watch: the curve doesn't cross there, it just touches . 4. Real Roots vs Complex Roots Not every root sits on the number line. The graph of f(x)=x^2+1 floats entirely above the x -axis — zero real roots. But FTA says a degree-2 polynomial has 2 roots, so where are they? In the complex plane: solving x^2=-1 gives . 5. Splitting Into Linear Factors

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