Parametric equations for roots

In summary, the conversation discusses the possibility of writing parametric equations for the graphs of polynomial equations, specifically in the case of plotting a real number 'b' on the x-axis and the real roots of the equation (7+b2)x3+(6-b)x2+9x-6=0 on the y-axis. The person suggests plotting a graph to inspire comments on the general situation and mentions the possibility of expressing roots as functions of coefficients.
  • #1
Big-Daddy
343
1
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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  • #2
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.
 

FAQ: Parametric equations for roots

What are parametric equations for roots?

Parametric equations for roots are mathematical equations that are used to find the roots or solutions of a polynomial or algebraic equation. These equations involve a variable parameter that takes on different values to find the corresponding roots.

How do parametric equations for roots work?

Parametric equations for roots use a substitution method to solve for the roots of a polynomial. This involves substituting a variable parameter into the equation and solving for the corresponding roots. This method is particularly useful for complex or higher degree polynomials.

What are the advantages of using parametric equations for roots?

Parametric equations for roots allow for a systematic and organized approach to finding solutions for polynomials. They also provide a more efficient way to find multiple roots and can handle complex or higher degree polynomials that may be difficult to solve by other methods.

Can parametric equations for roots be used for any type of polynomial?

Yes, parametric equations for roots can be used for any type of polynomial, including linear, quadratic, cubic, and higher degree polynomials. They can also be used for both real and complex roots.

Are there any limitations to using parametric equations for roots?

While parametric equations for roots can be useful for solving polynomials, they may not always provide the most accurate or precise solutions. Additionally, they may not work for every type of mathematical problem, such as systems of equations or inequalities.

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