What are the applications of roots of a polynomial?

[pairofstrings]

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.

 

[BvU]

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

 

[pairofstrings]

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?

 

[BvU]

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.

 

[symbolipoint]

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).

 

[pairofstrings]

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)