842 ALGEBRA.
Ex. 1. Find the fifth term of (a + 2x^3)^17,
Here (r + 1)= 5, therefore the required term =" Cia 2xr)-
_17-16.15.14 ~ 1-2-3-4 = 88080 ax,
Ex. 2. Find the fourteenth term of (8 — a),
Here r + 1 = 14, therefore
X 16 gByl2
the required term = 4 C43(3)2(— a)8 ="C x(—9al) [Art. 395.] =— 945 a8.
411. Simplest Form of the Binomial Theorem. The most convenient form of the binomial theorem is the expansion of (1+). This is obtained from the general formula of Art. 408, by writing 1 in the place of a, and @ in the place of 0. Thus
a 4h gy = 1 LOC tb "Op? + Bee Clete co ILC —1 =14n ee Oa par, n(n —1)(n — Z)---(n —7 +1) | SN a lz the general term being
412. The expansion of a binomial may always be made to depend upon the case in which the first term is unity; thus
(a+b)? = {4 + a} LS w(i +c)", where ¢ = 2 a
Ex. Find the coefficient of x^16 in the expansion of (x^2-2x)^10
We have (x^2-2x)^10 = x^20 (1 - 2 x)^10;
and, since x^20 multiplies every term in the expansion of (1 - 2 x)^10, we have in this expansion to seek the coefficient of the term which contains 1 x^4.
Hence the required coefficient = 10C4 (-2)^4 = 10 9 8 7 1 2 3 4 \times 16 = 3360.
