Roots of Related Cubics: kα, kβ, kγ

This example demonstrates how the roots of a cubic equation transform when each root α, β, γ is multiplied by a common factor k, showing the relationship between the original cubic and the new cubic with scaled roots. The core mathematical concepts include cubic equations, Vieta's formulas, linear transformations (scaling) of polynomial roots, and graphing polynomial functions. Construction details: using GeoGebra commands such as Point to create root points, Polynomial to define the polynomial, and LaTeX for displaying expressions; objects include k (slider), α, β, γ (original roots), d, c, b, f (coefficients or functions), text1–text8 (labels), ε, ζ, δ (scaled roots), p, q, r (related points/coefficients), and g (scaled polynomial). Suitable for high school or early college students to explore root transformations, and useful for inquiry-based learning or teacher demonstrations.