Binomial Theorem

Topics: Pascal's triangle, Binomial coefficient, Binomial theorem Pages: 4 (1409 words) Published: September 30, 2012

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of xn−kyk is equal to the number of different combinations of k elements that can be chosen from an n-element set.

HISTORY This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results. Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

STATEMENT OF THE THEOREM According to the theorem, it is possible to expand any power of x + y into a sum of the form where denotes the corresponding binomial coefficient. Using summation notation, the formula above can be written This formula is sometimes referred to as the binomial formula or the binomial identity. A variant of the binomial...
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