Finite Differences & Leading Coefficients Relation

AI Thread Summary
The discussion focuses on understanding the relationship between the nth finite difference of a polynomial and its leading coefficient. It is suggested to start by analyzing simple polynomial cases to observe patterns in their finite differences. The initial thought is that the sign of the nth finite difference corresponds to the sign of the leading coefficient. Participants are encouraged to explore specific examples, such as the first difference of a linear polynomial and higher-order differences for quadratic and cubic polynomials. This approach aims to clarify the concept and guide towards a solution.
dvmckay23
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Homework Statement


Determine the relation that exists between the nth finite difference and the leading coefficient.

Homework Equations


... I'm not too sure how to html it up properly, but the numbers/"n"s following the "a"s are meant to be sub-script:
f(x) = anx^n + an-1x^n-1 + ... + a2x^2 + a1x + a0

The Attempt at a Solution


If someone could even tell me the proper direction for the line of thought here. All I can figure out thus far is that the difference is the same sign (positive or negative) of the coefficient. Any hints would be GREATLY appreciated!

Thanks!
~D
 
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Are you assuming an nth degree polynomial?

Try some easy cases first. What is the first difference of ax+ b?
What is the second difference of ax^2+ bx+ c?
What is the third difference of ax^3+ bx^2+ cx+ d?
 

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