How do you use the binomial series to expand (x+1+x^-1)^4(x+1+x−1)4?
1 Answer
=x^4+4x^3+10x^2+16x+19+16x^(-1)+10x^(-2)+4x^(-3)+x^(-4)=x4+4x3+10x2+16x+19+16x−1+10x−2+4x−3+x−4
Explanation:
This is a trinomial, not a binomial. With binomials you could use Pascal's triangle to give you the coefficients.
Since the terms of the given trinomial are in geometric progression, we can use coefficients from a generalisation of Pascal's triangle that adds together
color(white)(00000000000000)1000000000000001
color(white)(00000000000)1color(white)(00)1color(white)(00)1000000000001001001
color(white)(00000000)1color(white)(00)2color(white)(00)3color(white)(00)2color(white)(00)1000000001002003002001
color(white)(00000)1color(white)(00)3color(white)(00)6color(white)(00)7color(white)(00)6color(white)(00)3color(white)(00)1000001003006007006003001
color(white)(00)1color(white)(00)4color(white)(0)10color(white)(0)16color(white)(0)19color(white)(0)16color(white)(0)10color(white)(00)4color(white)(00)1001004010016019016010004001
In this triangle each number is the sum of the three numbers above it: left, centre and right.
Hence we find:
(x+1+x^(-1))^4(x+1+x−1)4
=x^4+4x^3+10x^2+16x+19+16x^(-1)+10x^(-2)+4x^(-3)+x^(-4)=x4+4x3+10x2+16x+19+16x−1+10x−2+4x−3+x−4
