Infinite Degree Polynomials: Describing by Roots

In summary, the conversation discusses the possibility of describing infinite degree polynomials by their roots, similar to how finite degree polynomials are described. The speaker mentions that not all infinite degree polynomials have roots, but asks about polynomials like the power series of sin(x). The responder clarifies that there is no such thing as an infinite degree polynomial and suggests considering the analytic continuation of the power series, which has a general factorization theorem.
  • #1
foxjwill
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Homework Statement


Is it possible to describe some infinite degree polynomials by their roots in a way analagous to finite degree polynomials?


Homework Equations





The Attempt at a Solution



I know that, since not all infinite degree polynomials have roots (e.g. the power series representation of e^x), it would not be possible to do so for all of them. But what about polynomials like the power series of sin(x)? I was thinking maybe

[tex]\prod^\infty_{n=0} \left ( x^2 - n^2\pi^2 \right )[/tex]
 
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  • #2
Is it possible to describe some infinite degree polynomials...
There's no such thing as an infinite degree polynomial. I presume you mean a power series.

If (the analtyic continuation) of your power series is actually meromorphic, then there is a general factorization theorem. See:

http://en.wikipedia.org/wiki/Infinite_product
 

FAQ: Infinite Degree Polynomials: Describing by Roots

What are infinite degree polynomials?

Infinite degree polynomials are mathematical expressions that have an infinite number of terms, each with a variable raised to a different power. They can be written in the form of f(x) = a0 + a1x + a2x2 + a3x3 + ..., where a0, a1, a2, a3, ... are constants and x is the variable.

What is the degree of an infinite degree polynomial?

The degree of an infinite degree polynomial is equal to the highest power of the variable in the expression. For example, in the polynomial f(x) = 3x4 + 2x7 + x10, the degree is 10.

How are infinite degree polynomials useful?

Infinite degree polynomials are useful in many areas of mathematics, such as calculus, algebra, and geometry. They can be used to model real-world situations, solve equations, and make predictions. They also have important applications in physics, engineering, and economics.

What are the roots of an infinite degree polynomial?

The roots of an infinite degree polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the polynomial equation f(x) = 0. The number of roots is equal to the degree of the polynomial.

How can infinite degree polynomials be described by their roots?

Infinite degree polynomials can be described by their roots by factoring them into linear and quadratic factors. The roots of the polynomial are the values of the variable that make each factor equal to zero. By knowing the roots, we can determine the behavior of the polynomial, such as its end behavior and the location of its turning points. Additionally, knowing the roots can help us graph the polynomial and find its x-intercepts.

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