Applications of Polynomial Equation in Geometry

Certain geometric facts are most cleanly proved using polynomial equations — reducing a geometry question ("how many points can these two curves share?") to an algebra question ("how many roots can this polynomial equation have?").

Worked illustration (Example 3.14 — a line meets a circle in at most 2 points). Choose coordinate axes so the circle is $x^2+y^2=r^2$ and the line is $y=mx+c$. Points of intersection satisfy both simultaneously; substituting $y=mx+c$ into the circle's equation gives

a quadratic in $x$. A quadratic equation cannot have more than two roots — so a line and a circle cannot meet at more than two points. (One elegant side-observation: since $1+m^2,2mc,c^2-r^2$ are all real, this quadratic's two roots are either both real or a non-real conjugate pair — so the line and circle either meet at $2$ points, are tangent [one repeated real root], or don't meet at all [non-real roots]; a line can never touch a circle at exactly one, simple point without being tangent there.) …