Parametric equations for roots

AI Thread Summary
To plot the roots of the cubic equation (7+b^2)x^3+(6-b)x^2+9x-6=0 against a real number 'b' on the x-axis, parametric equations can be derived. The discussion suggests plotting the specific case when b=1 to visualize the relationship between 'b' and the roots. Participants express interest in whether polynomial roots can generally be expressed as functions of their coefficients. A suggestion is made to graph the curves for better understanding and to encourage further commentary on the topic. The exploration of polynomial roots as functions of coefficients remains a central theme in the discussion.
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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