High School What are the applications of roots of a polynomial?

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SUMMARY

The discussion centers on the applications of roots of quadratic polynomials, specifically the equations x² + 7x + 12 = 0 and x² + 8x + 12 = 0. The roots of these equations are identified as -3, -4, -6, and -2. Participants clarify that finding the roots is essential for determining intersection points of modeling equations, which can be represented graphically. The conclusion emphasizes that roots provide solutions to polynomial equations, allowing for the analysis of function behavior and intersections.

PREREQUISITES
  • Understanding of quadratic equations and their standard form.
  • Knowledge of polynomial roots and their significance in graphing.
  • Familiarity with the concept of function intersections.
  • Basic algebra skills, including solving equations.
NEXT STEPS
  • Study the graphical representation of quadratic functions and their roots.
  • Learn about polynomial long division and its applications in finding roots.
  • Explore systems of equations and methods for solving them, such as substitution and elimination.
  • Investigate the use of the Quadratic Formula for finding roots of quadratic equations.
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Students and educators in mathematics, particularly those focusing on algebra and polynomial functions, as well as anyone interested in understanding the practical applications of polynomial roots in modeling and graphing.

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.
 
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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)
 
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