Binomial Expansion long division

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SUMMARY

The discussion focuses on expanding the expression (1-2x+4x^2)^(-0.5) using the identity (1+2x)(1-2x+4x^2) = 1+8x^3. The correct expansion yields the series 1 + x - x^2/2 - 7x^3/2 + ... The participants clarify that the operation involves multiplication rather than division when applying the binomial expansion for (1+2x)^(0.5). The final solution confirms the correct approach to the problem.

PREREQUISITES
  • Understanding of binomial expansion
  • Familiarity with series expansion techniques
  • Knowledge of polynomial long division
  • Basic algebraic manipulation skills
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  • Study binomial expansion for fractional powers
  • Learn polynomial long division techniques
  • Explore series expansion of functions using Taylor series
  • Investigate the convergence of power series
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Students studying algebra, particularly those focusing on polynomial expansions and series, as well as educators looking for examples of binomial expansion applications.

chrisyuen
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Homework Statement



Using the identity (1+2x) (1-2x+4x^2) = 1+8x^3 to expand (1-2x+4x^2)^(-0.5) in ascending powers of x as far as the term in x^3. (The answer is 1+x-x^2/2-7x^3/2+...)

Homework Equations



(1+2x)^(0.5) = 1+x-x^2/2+x^3/2+...
(1+8x^3)^(-0.5) = 1-4x^3+...

The Attempt at a Solution



By long division, I have the solution of 1-x+3x^2/2-13x^3/2+...
 
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Since you have raised (1+2x) to the power of +(0.5), you should
be multiplying, not dividing. (Easy).
 
davieddy said:
Since you have raised (1+2x) to the power of +(0.5), you should
be multiplying, not dividing. (Easy).

Yes, I got it.

I forgot (1/(1+2x))^(-0.5) = (1+2x)^0.5.

Thank you very much!
 

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