binomial theorem

American  

noun

Mathematics.

1. the theorem giving the expansion of a binomial raised to any power.

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binomial theorem British  

noun

1. 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

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binomial theorem Scientific  

1. The theorem that specifies the expansion of any power of a binomial, that is, (a + b) ^m . According to the binomial theorem, the first term of the expansion is x ^m, the second term is 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 ^m is reached. The coefficient of x ^{m - r} is 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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Etymology

Origin of binomial theorem

First recorded in 1865–70

Example Sentences

Given his obsession with the binomial theorem, we based the code we created for him on Pascal’s triangle.

From Scientific American

Expand by the binomial theorem and simplify: 8.

From Project Gutenberg

The binomial theorem operates irrespective of the values substituted for its symbols.

From Project Gutenberg

Later still he made what seemed to be approaches toward Newton’s binomial theorem.

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