Page:Elementary algebra (1896).djvu/365

BINOMIAL THEOREM. 347 The greatest value of 7 consistent with this is 5; hence the greatest term is the sixth, and its value

418. Sum of the Coefficients. To jind the sum of the coefficients in the expansion of (1+x)n.

In the identity

(1+x)n=1+4 Ga + Ca? + Cr? + --- + Ca";

put x=1; thus 2=14+ G4 Q+++-+C, = sum of the coefficients.

Cor. Q4+O4+ G4+--4+¢,=27-1; that is, the total number of combinations of n things taking some or all of them at a time is 2n —1. [See Art. 400.]

419. Sums of Coefficients equal. To prove that in the expansion of (1+x)n, the sum of the coefficients of the odd terms is equal to the sum of the coefficients of the even terms.

In the identity

+a)" =1+ Cy + Cy? + Cy? + --- + C2’, put x=—1; thus 0=1-G4+6-G4+¢6,-C6,4+--; ~14646,4+--=O4+0,4+6,+---.

420. Expansion of Multinomials. The Binomial Theorem may also be applied to expand expressions which contain more than two terms.

Ex. Find the expansion of (x2+2x-1)3.

Regarding 2x-1 as a single term, the expansion

= (x2)3 + 3(x2)2(2x-1)+3x2(2a-1)2 +(2x-1)3 = x6 + 6x5 +9x4 -4x3 - 9x2+ 6x -1, on reduction.

421. Binomial Theorem for Negative or Fractional Index. For a full discussion of the Binomial Theorem when the index is not restricted to positive integral values the student