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Explain binomial theorem in high school mathematics in detail.
Let f (x) = (x 2+1) (x-9) = A0+a1(x-1)+A2 (x-1) 2+…+A1.

f'(x)=(x^2+ 1)'(x-9)+(x^2+ 1)(x-9)'=3x^2- 18x+ 1

=a 1+2a2(x- 1)+3a3(x- 1)^2+4a4(x- 1)^3+5a5(x- 1)^4+…+ 10a 10(x- 1)^9+ 1 1a 1 1(x- 1)^ 10

f'(- 1)=3*(- 1)^2- 18*(- 1)+ 1=22

= a 1-2 a2+3 a3-4a 4+…- 10a 10+ 1 1a 1 1 1

=(a 1+3 a3+5 a5+…+ 1 1a 1 1)-(2 a2+4 a4+6 a6+…+ 10a 10)

Therefore, (a1+3a3+5a5+…+11)-(2a2+4a4+6a6+…+10) = 22.