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.
(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.