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.

(iii.) Another result is that we can equate coefficients of like powers of x in two **multinomials** obtained from the same expression by different methods of expansion.

We therefore define algebraical division by means of algebraical multiplication, and say that, if P and M are **multinomials**, the statement " P/M = Q " means that Q is a multinomial such that MQ (or QM) and P are identical.