Proving Polynomial Property: Get Hint Here

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SUMMARY

The discussion centers on proving that for each real number a, the function p defined by p(x) = f(x+a) is a polynomial of degree n. A critical insight is that the function f must meet specific conditions to ensure that p is indeed a polynomial; without these conditions, such as when f(x) = sin(x), p does not qualify as a polynomial. Participants emphasize the importance of understanding the definitions and properties of the function f to approach the proof correctly.

PREREQUISITES
  • Understanding of polynomial functions and their properties
  • Familiarity with function transformations, specifically translations
  • Knowledge of real analysis concepts related to continuity and differentiability
  • Basic understanding of trigonometric functions and their properties
NEXT STEPS
  • Review the definitions and properties of polynomial functions
  • Study function transformations, focusing on horizontal shifts
  • Explore conditions under which a function is classified as a polynomial
  • Investigate examples of non-polynomial functions, such as trigonometric functions
USEFUL FOR

Students studying real analysis, mathematicians interested in polynomial properties, and educators seeking to clarify function classifications in mathematical proofs.

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Homework Statement



Prove the following

For each real a, the function p given by p(x) = f(x+a) is a polynomial of degree n.

Homework Equations


Can I have a hint I have hard time starting.Thanks


The Attempt at a Solution

 
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Jbreezy said:

Homework Statement



Prove the following

For each real a, the function p given by p(x) = f(x+a) is a polynomial of degree n.

Homework Equations


Can I have a hint I have hard time starting.Thanks


The Attempt at a Solution


I think the hint would be to re-read the question and any material that is being assumed as background to it, as there has to be some condition or definition on f else it is not even true. That condition would be what you have to use.
 
For example, if f(x)= sin(x), then p(x)= sin(x+ a) is NOT a polynomial. Surely, you have misunderstood the question.
 

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