binomial bī˝nō´mēəl [key], polynomial expression (see polynomial ) containing two terms, for example, x + y. The binomial theorem, or binomial formula, gives the expansion of the n th power of a binomial ( x + y ) for n =1, 2, 3, … , as follows: where the ellipsis (…) indicates a continuation of terms following the same pattern. For example, using the formula and reducing fractions, one obtains ( x + y ) 5= x 5+5 x 4 y +10 x 3 y 2+10 x 2 y 3+5 x y 4+ y 5. The coefficients 1, n, n ( n −1)/1·2, etc., of x and y may also be found from an array known as Pascal's triangle (for Blaise Pascal ), formed by adding adjacent numbers to find the number below them as follows:
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