University of South AfricaMAT 1581

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A mathematical expression which consists of only two terms, say a + b, is called a binomial (“bi” means two). In this module we are investigating the powers of a binomial, that is (a + b)n. The bi... nomial theorem gives us a quick way to raise an expression comprising two terms to any given power. This theorem is used to work out annuity formulae in financial management and differential formulae in this course. Your pocket calculator uses this theorem in its calculations, for example to extract roots. If n in (a + b)n is small, we can easily use multiplication to expand the series, but if n becomes bigger, say (a + b)20, multiplication becomes tedious. Examine the following expansions: a  b2  a  ba  b  a2  2ab  b2 a  b3  a  ba  ba  b  a3  3a2b  3ab2  b3 a  b4  a  ba  ba  ba  b  a4  4a3b  6a2b2  4ab3  b4 We can conclude that a  b5 would begin with a5 and end with b5 a  b20 a  bn would begin with a20 and end with b20 would begin with an and end with bn Note this: a  b2 a  b3 a  b4 expands to three terms expands to four terms expands to five terms We can conclude that a  b5 would expand to six terms a  b20 a  bn would expand to twenty-one terms would expand to (n + 1) terms We notice that • the coefficients read the same backwards as forwards • the first coefficient (and the last one) is 1 • in all the expansions the powers of a are descending and the powers of b are ascending • the sum of the indices of a and b is n [Show More]

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