If we expand the symbolic expression by the **multinomial** theorem, and remember that any symbolic product ai 1 a2 2 a3 3 ...

A **multinomial** consisting of two or of three terms is a binomial or a trinomial.

Now log (1+ï¿½X1 +/22X2+/ï¿½3X3 +ï¿½ï¿½ï¿½) =E log (1+/2aix1+22aix2-1-/23ax3+...) whence, expanding by the exponential and **multinomial** theorems, a comparison of the coefficients of ï¿½n gives (n) (-)v1+v2+v3+..

By the exponential and **multinomial** theorems we obtain the results) 1,r -1 (E7r) !

An expression denoting that two or more monomials are to be added or subtracted is a **multinomial** or polynomial, each of the monomials being a term of it.

(viii.) The quadratic equation is the equation of two expressions, monomial or **multinomial**, none of the terms involving any power of x except x and x 2 .

A°a n in the n+ I terms of the **multinomial** equivalent to (A+a)".

In the same way we have (A-a) 2 =A 2 -2Aa+a 2, (A-a)3 = A 3 -3A 2 a+3Aa 2 -a 3, ..., so that the **multinomial** equivalent to (A-a)" has the same coefficients as the **multinomial** equivalent to (A+a)", but with signs alternately + and -.

The **multinomial** which is equivalent to (A= a)", and has its terms arranged in ascending powers of a, is called the expansion of (A= a) n.

We know that (A+a)" is equal to a **multinomial** of n+I terms with unknown coefficients, and we require to find these coefficients.

- (i.) If we divide the **multinomial** P=p0xn+plxn-1+...

If, moreover, we examine the process of algebraical division as illustrated in ï¿½ 50, we shall find that, just as arithmetical division is really the solution of an equation (ï¿½ 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so algebraical division by a **multinomial** really implies the use of undetermined coefficients (ï¿½ 42).

The terms trinomial, quadrinomial, **multinomial**, &c., are applied to expressions composed similarly of three, four or many quantities.

He introduced the terms **multinomial**, trinomial, quadrinomial, &c., and considerably simplified the notation for decimals.

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.