Fundamental Theorem Preview

Algebra 2 · Axiom Academy

LESSON The Fundamental Theorem, Previewed Count complex numbers and repeated roots, and every quadratic has exactly two roots — a first look at the Fundamental Theorem of Algebra. Start with a quadratic whose parabola crosses the x-axis twice. Each crossing is a real root, and the two linear factors say exactly where. 2. No Real Roots — but Still Two Now take a parabola that floats entirely above the x-axis. It never crosses, so there are no real roots . Yet the quadratic formula still returns two answers — they simply aren't real. The discriminant , so the square root is imaginary and the two roots are the complex numbers 1+2i and 1-2i . What about a parabola that just touches the x-axis at one point? It looks like a single solution — but the factored form shows the same factor twice. 4. Complex Roots Come in Pairs Look again at the complex roots . On the complex plane they are mirror images across the real axis — a conjugate pair . 5. The Pattern: The Fundamental Theorem Every case landed on the same count. A degree-2 polynomial always breaks into exactly two linear factors — so it always has exactly two roots, once we count in the right number system and count repeats. Every polynomial of degree — with real or complex coefficients — has exactly n roots in the complex numbers , counted with multiplicity . You've seen why a quadratic always has exactly two roots — and previewed the theorem that a degree- n polynomial has exactly n .

This is the written version of the interactive lesson above. See the full Algebra 2 course.