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Question

Mathematics Question on Binomial Theorem for Positive Integral Indices

Expand the expression (x+1x)6(x+ \frac{1}{x})^6.

Answer

By using Binomial Theorem, the expression (x+1x)6(x+ \frac{1}{x})^6 can be expanded as

(x+1x)6(x+ \frac{1}{x})^6 = ^6C_0 (x)^6 + ^6C_1(x)^5(\frac{1}{x}) \+ ^6C_2(x)^4(\frac{1}{x}) \+ ^6C_3(x)^3(\frac{1}{x})^3 + ^6C_4(x)^2(\frac{1}{x})^4 +$$ ^6C_5(x)(\frac{1}{x})^5 + ^6C_6(\frac{1}{x})^6

=x6+6(x)5(1x)+15(x)4(1x2)+20(x)3(1x3)+15(x)2(1x4)+6(x)(1x5)+1x6x^6 + 6(x)^5(\frac{1}{x}) + 15(x)^4(\frac{1}{x^2}) + 20(x)^3(\frac{1}{x^3}) + 15 (x)^2 (\frac{1}{x^4}) + 6(x)(\frac{1}{x^5}) + \frac{1}{ x^6}

=x6+6x4+15x2+20+15x2+6x4+1x6x^6 + 6x^4 + 15x^2 + 20+ \frac{15}{x^2} + \frac{6}{x^4} + \frac{1}{ x^6}