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# binomial theorem

## noun

,

*Mathematics.*- the theorem giving the expansion of a binomial raised to any power.

binomial theorem

## noun

- a mathematical theorem that gives the expansion of any binomial raised to a positive integral power,
*n*. It contains*n*+ 1 terms: (*x*+*a*)*n*=*xn*+*nx*^{n}^{–}^{1}*a*+ [*n*(*n*–1)/2]*xn*^{–}²*a*² +…+ (*nk*)*xn*^{–}*kak*+ … +*an*, where (*nk*) =*n*!/(*n–k*)!*k*!, the number of combinations of*k*items selected from*n*

binomial theorem

- The theorem that specifies the expansion of any power of a binomial, that is, (
*a*+*b*)^{m}.*x*, the second term is^{m}*mx*^{m-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-r}*m*![*r*!(*m*−*r*)!]. Thus the expansion of (*a*+*b*)^{3}is*a*^{3}+ 3*a*^{2}*b*+ 3*ab*^{2}+*b*^{3}.

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## Word History and Origins

Origin of binomial theorem^{1}

First recorded in 1865–70

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## Example Sentences

You might as well try to rush the Proof of the Binomial Theorem.

From Project Gutenberg

A chapter catches my attention in the middle of the volume; it is headed, Newton's Binomial Theorem.

From Project Gutenberg

What can a binomial theorem be, especially one whose author is Newton, the great English mathematician who weighed the worlds?

From Project Gutenberg

Expand each term by the binomial theorem, and let us fix our attention on the coefficient of yn−1.

From Project Gutenberg

Why, as far back as when I was studying algebra, I nobly refused to learn the binomial theorem.

From Project Gutenberg

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