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) !
If we expand the symbolic expression by the multinomial theorem, and remember that any symbolic product ai 1 a2 2 a3 3 ...
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.
A multinomial consisting of two or of three terms is a binomial or a trinomial.
(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.
The total number of such terms in the multinomial equivalent to (A+a) (B+b) (C+c) (D+d) (E+e) is therefore (3.4.
We know that (A+a)" is equal to a multinomial of n+I terms with unknown coefficients, and we require to find these coefficients.
The use of negative coefficients leads to a difference between arithmetical division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms. The most important case is division by a binomial, as illustrated by the following examples: - 2.10+1) 6.100+5.10+ 1(3.10+I 2.10+I) 6.100+I.10 - I (3.10 - I 6.100+3.10 6.100+3.10 2.10+ I - 2.10 - I 2.10 +I - 2.10 - I In (1) the division is both arithmetical and algebraical, while in (2) it is algebraical, the quotient for arithmetical division being 2.10+9.
- (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).
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.
He introduced the terms multinomial, trinomial, quadrinomial, &c., and considerably simplified the notation for decimals.
The terms trinomial, quadrinomial, multinomial, &c., are applied to expressions composed similarly of three, four or many quantities.
The multi- (or poly-) nomial theorem has for its object the expansion of any power of a multinomial and was discussed in 1697 by Abraham Demoivre (see Combinatorial Analysis).
(1) Ile /3 c Tou irvpiov, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Hepi On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) `H Ka06Xov lrpa-yµareta, perhaps a work on the general principles of mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) ' (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of 7r than the 37 and 3,4-A of Archimedes; (6) an arithmetical work (as to which see Pappus) on a system of expressing large numbers in language closer to that of common life than that of Archimedes' Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm.