The extension to multinomials forms part of the theory of factors (ï¿½ 51).
(v.) When we have to multiply two multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication.
Continuing to develop the successive powers of A+a into multinomials, we find that (A+a)3=A3+3A2a+3Aa2+a3, &c.; each power containing one more term than the preceding power, and the coefficients, when the terms are arranged in descending powers of A, being given by the following table I I ' 'I 2 I 1 3 3 I 4 6 4 I 5 IO to 5 I I x 6 15 20 15 6 &c., where the first line stands for (A+a)°=1.
(vi.) It follows that, if two multinomials of the nth degree in x have equal values for more than n values of x, the corresponding coefficients are equal, so that the multinomials are equal for all values of x.
- (i.) The results of the addition, subtraction and multiplication of multinomials (including monomials as a particular case) are subject to certain laws which correspond with the laws of arithmetic (ï¿½ 26 (i.)) but differ from them in relating, not to arithmetical value, but to algebraic form.