Algebra: Expressibility of f(x) with 10 Numbers

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In summary, the conversation discusses the possibility of expressing a function of a series of numbers as a polynomial equation, either in one or two variables, and the use of Lagrange's interpolation formula or Newton's divided difference scheme to find it. It is also mentioned that while a finite sequence of points can be represented as a polynomial in two variables, it may not necessarily be expressible as a product of two polynomials.
  • #1
sid_galt
502
1
Let's say you have ten numbers

[tex]
f(1) = 1
[/tex]
[tex]
f(2) = 100
[/tex]
[tex]
f(3) = 45
[/tex]
[tex]
f(4) = 9000
[/tex]
[tex]
f(5) = 999
[/tex]
[tex]
f(6) = 46
[/tex]
[tex]
f(7) = 47
[/tex]
[tex]
f(8) = 48
[/tex]
[tex]
f(9) = 59
[/tex]
[tex]
f(10) = 60
[/tex]

Is f(x) expressible in the form

[tex]
f(x)=a_nx^n+a_{n-1}x^{n-1}...a_1x+a_0
[/tex]
or perhaps
[tex]
f(x)=(a_nx^n+a_{n-1}x^{n-1}...a_1x+a_0)(b_ny^n+b_{n-1}y^{n-1}...b_1y+b_0)
[/tex]
Why? Why not?

If it is, is there any way to find it?
 
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  • #2
There are an infinite number of answers to your homework.
 
  • #3
It's not homework :smile:

matt grime said:
There are an infinite number of answers to your homework.

So can any function of a random series of numbers be expressed as a product of two or more polynomial equations if f is a function of two variables or one polynomial equation if f is a function of one variable?
 
Last edited:
  • #4
i don't really know the answer but i think "lagrange interpolation" might have something to do with this.
 
  • #5
Firstly, you shouldn't have an input x into f(x) and an output in two variables.

And of course given a finite number of points x, f(x) there are an infinite number of polynomials through those points.
 
  • #6
Given any finite number, n, of points (x, y) there exist an infinite number of functions (and polynomials) whose graphs pass through those points (i.e. y= f(x)).

However, there exist a unique polynomial of degree n+1 (or lower if the points are not "independent") whose graph passes through those points.

As Fourier jr. said, Lagrange's interpolation formula will give that polynomial. Newton's divided difference scheme will also work.

A finite sequence of points (x, y, z) CAN be represented as a polynomial in the two variables (x,y). However, I do not believe that it can necessarily be represented as a polynomial in x TIMES a polynomial in y.
 
  • #7
Not quite true. The x's corresponding to distinct y's must be distinct (ie. if [itex](x_n, y_n), \ (x_m, y_m)[/itex] are some of the points and [itex]x_n = x_m[/itex] then in order to have a set [itex](x, \ f(x))[/itex] for a polynomial [itex]f(x)[/itex] containing both points you need [itex]y_n=y_m[/itex]), then it's fine~
 

Related to Algebra: Expressibility of f(x) with 10 Numbers

1. What is the purpose of expressing f(x) with 10 numbers in Algebra?

The purpose of expressing f(x) with 10 numbers in Algebra is to simplify complex mathematical equations and make them easier to solve.2. How do you determine which 10 numbers to use for f(x)?

The 10 numbers used for f(x) are typically determined by the function itself and its corresponding variables. These numbers can also be chosen based on the problem being solved.3. Can f(x) be expressed with less than 10 numbers?

Yes, f(x) can sometimes be expressed with fewer than 10 numbers. This depends on the complexity of the function and the variables involved.4. Why is it important to understand the expressibility of f(x) with 10 numbers?

Understanding the expressibility of f(x) with 10 numbers is important because it allows for efficient problem-solving and a deeper understanding of algebraic concepts.5. Are there any limitations to expressing f(x) with 10 numbers?

Yes, there are limitations to expressing f(x) with 10 numbers. Some functions may require more numbers to accurately represent the function and its variables. Additionally, the accuracy of the solution may depend on the chosen 10 numbers.

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