# Binomial-theorem meaning

*a*+

*b*)

*of a binomial (*

^{m}*a*+

*b*) as a certain sum of products

*a*, such as (

^{i}b^{j}*a*+

*b*)

^{2}=

*a*

^{2}+ 2

*ab*+

*b*

^{2}.

^{n}: discovered by Omar Khayyám and generalized by Sir Isaac Newton (Ex.: (a + b)

^{2}= a

^{2}+ 2ab + b

^{2})

*a*+

*b*)

*According to the binomial theorem, the first term of the expansion is*

^{m}.*x*, the second term is

^{m}*mx*

^{m&spminus;1}

*y,*and for each additional term the power of

*x*decreases by 1 while the power of

*y*increases by 1, until the last term

*y*is reached. The coefficient of

^{m}*x*is

^{m&spminus;r}*m*![

*r*!(

*m*−

*r*)!]. Thus the expansion of (

*a*+

*b*)

^{3}is

*a*

^{3}+ 3

*a*

^{2}

*b*+ 3

*ab*

^{2}+

*b*

^{3}.