Algebra Binomial Theorem and Binomial Series Draft for Information Only
Content Binomial Theorem and Binomial Series[]
Binomial Theorem and Binomial Series[]A binomial is an algebraic polynomial expression of form (a+b). The binomial theorem is a theorem descibing the algebraic expansion of the powers of a binomial into an infinite binomial series. Binomial Expansion[]The expansion of the powers of a binomial can be obtained by direct multiplication. Imply The number of terms of the binomial expansion is alway equal to the power of the binomial plus 1. The exponent of a decreases by 1 from the power of the binomial while the exponent of b increases by 1 from zero or vice versa. Therefore the first term is a to the power of the binomial and the last term is b to the power of the binomial, and the sum of the exponents of a and b is always equal to the power of the binomial. The coefficient of a is a specific integer depending on the power of the binomial and the exponent of a or b or vice versa. The coefficient of each term can be obtained by multiplying the coefficient of previous term by the exponent of a in the previous term and dividing the product by the item number of the previous term instead of direct multiplaction. Pascal's Triangle[]The coefficients of powers of binomial, (a+b) can be arranged in the format of Pascal triangle with the first coefficient of the powers of binomial on the same row or column. The coefficients of powers of binomial can also be arranged in the an "equilateral" triangle pattern of same numbers of coefficients on the edge. Imply
The row number on the left is equal to the power of the binomial with the expanded binomial on the left. Similarly, the exponent of a decreases by 1 from the power of the binomial to zero while the exponent of b increases by 1 from zero to the power of the binomial or vice versa. The number of term in each row is equal to the power of the binomial plus. And the sum of the coefficient of the binomial expansion in each row is equal to 2 to the power of the power of the binomial. With this rearrangement, the coefficient of each term is equal to the sum of the two coefficients of the row above immediately. The coefficients of powers of binomial can therefore be determined by the pattern of a Pascal's triangle, rather than direct multiplication. Properties of Pascal's Triangle
Binomial Theorem[]The binomial theorem is developed to expand binomials of any given power directly by describing the algebraic expression of the general form of a binomial expansion. The binomial theorem states that the binomial (a+x) to the power of a nonnegative integer n can be expressed as If a=1, then the general binomial series can be reduced to Binomial Series[]In fact, the number n in the binomial theorem can be extended to any number. For any number n, the expansion of the binomial should be expressed in form of an infinite series, called binomial series. For any number n, if the absolute value of x is less than one, then the series converges absolutely. If n is a nonnegative integer, the infinite binomial series is reduced to a finite series having the same form of the binomial theorem ©sideway ID: 130500022 Last Updated: 22/3/2016 Revision: 1 Ref: References
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