What are the applications of roots of a polynomial?

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Discussion Overview

The discussion revolves around the applications of the roots of polynomials, specifically quadratic equations. Participants explore how roots can be used to find intersections between modeling equations and the implications of these intersections in various contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that finding the roots of polynomials can help determine where two modeling equations intersect, indicating a potential application in understanding differences between models.
  • Another participant proposes sketching the parabolas to visualize the intersection points, emphasizing the importance of finding where the quadratic equations intersect the x-axis.
  • A later reply questions the clarity of the term 'finding roots of a system of polynomials' and distinguishes between roots of a single polynomial and solutions to a system of equations.
  • Another participant clarifies that the discussion seems to be about finding where the functions of two quadratic equations intersect, providing a method to solve for intersection points.
  • One participant concludes that the root or zero of an equation represents the answer to the question posed by the function, reinforcing the concept of roots as solutions.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of roots in the context of systems of equations. While some focus on the intersection of functions, others challenge the terminology and clarity of the discussion. No consensus is reached on the applications of roots beyond the examples provided.

Contextual Notes

There are limitations in the discussion regarding the definitions of terms like 'roots of a system of polynomials' and the assumptions underlying the intersection of functions. The mathematical steps to arrive at conclusions are not fully resolved.

Who May Find This Useful

This discussion may be useful for individuals interested in the applications of polynomial roots in modeling, those studying algebra, or anyone exploring the intersection of mathematical functions.

pairofstrings
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Hello.

Assume that I have two polynomials of degree 2, i.e., Quadratic Equations.

1.
Assume that the Quadratic Equation is:
x2 + 7x + 12 = 0
The roots of the Quadratic Equation is -3 and -4.

2.
Assume that there is another Quadratic Equation:
x2 + 8x + 12 = 0
The roots of the Quadratic Equation is -6 and -2.

Then the use of the roots of the polynomial is that:
When I am trying to find where two modeling equations intersect, where information overlaps, this is equivalent to finding the zeroes to know the difference of the models.
The modeling equations I chose to consider is delineated as 1 and 2 above.

What I think is, I find roots of two or more polynomials to know the differences between them or to do something else, like, initiating another curve from any of the roots.
Am I right?
I want to know the applications of a root of a polynomial.
Do you have an example which illustrates use of a root of a polynomial?

Thank you.
 
Last edited:
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Why don't you draw (sketch) the two parabolas ?

Solving one of the quadratic equations gives you the (0, 1 or 2) points where they intersect the x-axis (i.e. where the quadratic form has value 0).

Solving for the difference = 0 gives you possible intersection points. In this case the difference of the two forms reads x = 0
 
BvU said:
In this case the difference of the two forms reads x = 0
From this statement, I think you confirmed that finding roots of a system of polynomials means finding the differences between those polynomials with respect to other polynomial.

Right?
 
The expression 'roots of polynomials' seems to confuse you.

I don't know what you mean with 'finding roots of a system of polynomials'.

What I do know is that you can try to find solutions for a system of equations.

And I know what the roots of a single polynomial are.
 
pairofstrings

You know something about roots of polynomial (or at least for quadratic) equations. You give a description more in line with finding the solution of a SYSTEM of quadratic equations.
You are asking in effect, given x^2+7x+12=0 and x^2+8x+12=0, where do the FUNCTIONS which the left-hand members intersect?

If that is what you are asking, then x^2+7x+12=x^2+8x+12.
You can solve this. Subtract x^2 and 12, from both sides.
7x=8x
0=8x-7x
0=x

The two FUNCTIONS would seem to intersect at point (0,12).

Keep studying and this will become clear (if not today, then sometime during Intermediate Algebra).
 
symbolipoint said:
pairofstrings

You know something about roots of polynomial (or at least for quadratic) equations. You give a description more in line with finding the solution of a SYSTEM of quadratic equations.
You are asking in effect, given x^2+7x+12=0 and x^2+8x+12=0, where do the FUNCTIONS which the left-hand members intersect?

If that is what you are asking, then x^2+7x+12=x^2+8x+12.
You can solve this. Subtract x^2 and 12, from both sides.
7x=8x
0=8x-7x
0=x

The two FUNCTIONS would seem to intersect at point (0,12).

Keep studying and this will become clear (if not today, then sometime during Intermediate Algebra).

Thanks for this information.
I came to an answer to my original post.

The root or zero or solution of an equation is the answer to the question.
I find root, to get the answer to the question.

'y' is a function in 'x'.
y = f(x)

So, to know 'x', I should make 'y' as nothing or zero.
0 = f(x)
 
Last edited:

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