Binomial Theorem For Quadratic Equation

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SUMMARY

The discussion focuses on finding the coefficient of x^5 in the expression (1+x+x^2)^4 using the binomial theorem. The user expresses difficulty in applying the theorem to expressions involving terms like x^2. A suggested approach is to redefine the problem by letting b = x + x^2 and expanding b^n to isolate the terms contributing to x^5. This method effectively simplifies the expansion process and allows for the identification of the desired coefficient.

PREREQUISITES
  • Understanding of the binomial theorem and its applications.
  • Familiarity with polynomial expansion techniques.
  • Basic algebraic manipulation skills.
  • Knowledge of combinatorial coefficients.
NEXT STEPS
  • Study the binomial theorem in detail, particularly for expressions of the form (1 + ax + bx^2).
  • Practice polynomial expansions using different values of n and varying coefficients.
  • Explore combinatorial methods for finding coefficients in polynomial expansions.
  • Learn about generating functions and their applications in combinatorial problems.
USEFUL FOR

Students studying algebra, mathematicians interested in combinatorial techniques, and educators looking for effective methods to teach polynomial expansions.

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Question:
Find the coefficient of x^5 in (1+x+x^2)^4.


Problem:
I have not come across expanding brackets which have x^2. I know how to apply the binomial theorem for (a+b)^n or (1+a)^n but have not come across (1 + ax + ax^2)^n. They are not explained in my textbooks so I was wondering if you could provide hints or redirect me to a useful link. Thanks.
 
Physics news on Phys.org
Just use b=x+x^2 and if you need to expand out b^2,b^3,b^4 just use the terms that will give you x^5
 

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