Parametric equations for roots

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SUMMARY

The discussion centers on deriving parametric equations for the roots of the polynomial equation (7+b²)x³ + (6-b)x² + 9x - 6 = 0, where 'b' is a real number plotted on the x-axis and the corresponding real roots on the y-axis. An example is provided with b=1, yielding a root of approximately 0.46124674. The conversation suggests that visualizing the graph of these roots may lead to further insights into the relationship between polynomial coefficients and their roots.

PREREQUISITES
  • Understanding of polynomial equations and their roots
  • Familiarity with parametric equations
  • Basic graphing skills using mathematical software
  • Knowledge of cubic functions and their properties
NEXT STEPS
  • Explore how to derive parametric equations for polynomial roots
  • Learn to use graphing tools like Desmos or GeoGebra for visualizing polynomial functions
  • Investigate the relationship between polynomial coefficients and their roots using Vieta's formulas
  • Study numerical methods for finding roots of cubic equations, such as Newton's method
USEFUL FOR

Mathematicians, students studying algebra and calculus, and anyone interested in the graphical representation of polynomial roots and their dependence on coefficients.

Big-Daddy
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Can I write the parametric equations for the graphs in the following case:

on the x-axis, I want to plot a real number 'b'. On the y-axis, I want to plot the roots (all real roots) for x of the equation (7+b2)x3+(6-b)x2+9x-6=0. e.g. when b=1, I plot 1 on the x-axis and x=0.46124674 (the real root of the equation 8x^3+5x^2+9x-6=0) on the y-axis.

(I chose the coefficients on the cubic pretty much arbitrarily but I want to see how it's done.)
 
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Big-Daddy said:
Can I write the parametric equations for the graphs in the following case:

I suggest that you plot a graph of the curves involved in that example and post it. Maybe that will inspire someone to comment on the general situation.

I'm guessing that your are interested in the general question of whether the roots of a polynomial equation can be expressed as functions of the coefficients of the equation.
 

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