Can Taylor's Series Help Expand Functions in Powers of x-1?

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Homework Help Overview

The problem involves expanding the function f(x) = x^3 + 3x^2 + 15x - 10 in powers of x - 1, utilizing Taylor's series or other methods of polynomial expansion.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to start the expansion and seeks guidance on applying Taylor's series. Some participants suggest using the Taylor series formula directly, while others propose a substitution method to simplify the polynomial before expansion.

Discussion Status

Participants are exploring different methods for the expansion, including direct application of Taylor's series and a substitution approach. There is no explicit consensus, but multiple avenues for tackling the problem are being discussed.

Contextual Notes

The original poster mentions an impending exam, indicating a time constraint that may influence their approach to the problem. There is also a note about the polynomial having only four relevant derivatives, which may affect the method chosen for expansion.

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Homework Statement


Expand f(x)=x^3 + 3x^2 +15x -10 in powers of x-1


Homework Equations



Taylor's Series, Maclaurin's series

The Attempt at a Solution



I don't know how to start...I do have an idea of both the theorems but don't know how to apply it to this situation.

please help...sem exams in a week's time!
 
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x-1 means centered about x=1; the polynomial has only 4 relevant derivatives so just use the formula
 


One way to do this, without using Taylor's series, is to let u= x-1 so that x= u+1. Your polynomial becomes (u+1)^3 + 3(u+1)^2 +15(u+1) -10. Expand that in powers of u and, finally, replace u by (x-1) to get the polynomial in powers of x-1.

But simpler is to use the Taylor's series formula:
f(a)+ f`(a)(x-a)+ f``(a)/2(x-a)^2+ f```(a)/3! (x-a)^3+ ...
 


thanks
 

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