174 ALGEBRA.
In these results, spoken of as expansions, we notice that:
(1) The number of terms equals the index of the binomial plus one.
(2) The exponent of a in the first term is the same as the index of the binomial, and decreases by one in each succeeding term.
(3) The quantity b appears for the first time in the second term of the expansion with an exponent 1, and its exponent increases by one in each succeeding term.
(4) The coefficient of the first term is 1.
(5) The coefficient of the second term is the same as the index of the binomial.
(6) The coefficient of any term may be found by multiplying the coefficient of the preceding term by the exponent of a in that term, and dividing the result by the exponent of b plus 1.
Ex. 1. Expand (a + hy.
(a + by = a'^ + a^ + a^ + «3 + «2 + a + & +6-2 + 63 + M + 65 + 66 Coefficients, 1 + 6 +15 + 20 + 15 + 6 + 1 Multiplying, x^ -]- 6 w>h + 15 d^h"^ + 20 a^h^ + 15 cfih^ + 6 ah^ + 6^
Ex. 2. Expand (a - 2 62)*. (a_26'0*=[«+(-262)]* = a* + a3 + ^2 + a + (_2 62) + (-2 62)2+ (-2 62)3+ (-2 62)4 Coefficients, 1+4 +6 +_4 +J Multiplying, a'^-'^a^b +24a2&4 _32a66 +16a8
Note. The student will observe that in the line of coefficients, terms at equal distances from the beginning and the end are equal.
194. The same method may be used in expanding any multinomial.
Ex. Expand (a + 2 6 - c)^. (a + 26-c)3=[(a + 2 6) + (-c)]3 = (a + 2 6)3 + (a + 2 6)2 +(a + 26) + (-c) +(-c)2 +(-c)3 Coefficients, 1 +3 +3 + 1 Multiplying, (a + 2 6)3 - 3(rt + 2 6)2c + 3(a + 2 6)c2 - c^
