dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (§ 28 (iv.)), and are arranged in descending or ascending powers of x. If P and M are rational integral functions of x, arranged in descending powers of x, the division of P by M is complete when we obtain a remainder R whose degree (§ 45) is less than that of M. If R＝0, then M is said to be a factor of P.
The highest common factor (or common factor of highest degree) of two rational integral functions of x is therefore found in the same way as the G.C.M. in arithmetic; numerical coefficients of the factor as a whole being ignored (cf. § 36 (iv.)).
52. Relation between Roots and Factors.—
(i.) If we divide the multinomial
P ≡ p0xn+p1xn−1+...+pn
by x−a, according to algebraical division, the remainder is
R ≡ p0an+p1an−1+...+pn.
This is the remainder-theorem; it may be proved by induction.
(ii.) If x＝a satisfies the equation P＝0, then p0an+p1an−1+...+pn＝0; and therefore the remainder when P is divided by x−a is 0, i.e. x−a is a factor of P.
(iii.) Conversely, if x−a is a factor of P, then p0an+p1an−1+...+pn＝0; i.e. x＝a satisfies the equation P＝0.
(iv.) Thus the problems of determining the roots of an equation P＝0 and of finding the factors of P, when P is a rational integral function of x, are the same.
(v.) In particular, the equation P＝0, where P has the value in (i.), cannot have more than n different roots.
The consideration of cases where two roots are equal belongs to the theory of equations (see Equation).
(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.
53. Negative Indices and Logarithms.—(i.) Applying the general principles of §§ 47-49 to indices, we find that we can interpret X−m as being such that
Xm.X−m＝X0＝1; i.e. X−m＝1/Xm.
In the same way we interpret X−p/q as meaning 1/Xp/q.
(ii.) This leads to negative logarithms (see Logarithm).
54. Laws of Algebraic Form.—(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 (§ 26(i.)) but differ from them in relating, not to arithmetical value, but to algebraic form. The commutative law in arithmetic, for instance, states that a+b and b+a, or ab and ba, are equal. The corresponding law of form regards a+b and b+a, or ab and ba, as being not only equal but identical (cf. § 37 (ii.)), and then says that A+B and B+A, or AB and BA, are identical, where A and B are any multinomials. Thus a(b+c) and (b+c)a give the same result, though it may be written in various ways, such as ab+ac, ca+ab, &c. In the same way the associative law is that A(BC) and (AB)C give the same formal result.
These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in § 52 (vi.).
(ii.) One result of these laws is that, when we have obtained any formula involving a letter a, we can replace a by a multinomial. For instance, having found that (x+a)2＝x2+2ax+a2, we can deduce that (x+b+c)2＝{x+(b+c)}2＝x2+2(b+c)x+(b+c)2.
(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. For instance, by equating coefficients of xr in the expansions of (1 + x)m+n and of (1+x)m.(1+x)n we obtain (22) of § 44 (ii.).
(iv.) On the other hand, the method of equating coefficients often applies without the assumption of these laws. In § 41 (ii.), for instance, the coefficient of An−rar in the expansion of (A+a)(A+a)n−1 has been called (nr); and it has then been shown that (nr)＝(n−1r) + (n−1r−1). This does not involve any assumption of the identity of results obtained in different ways; for the expansions of (A+a)2, (A+a)3, . . . are there supposed to be obtained in one way only, viz. by successive multiplications by A+a.
55. Algebraical Division.—In order to extend these laws so as to include division, we need a definition of algebraical division. The divisions in §§ 50-52 have been supposed to be performed by a process similar to the process of arithmetical division, viz. by a series of subtractions. This latter process, however, is itself based on a definition of division in terms of multiplication (§§ 15, 16). If, moreover, we examine the process of algebraical division as illustrated in § 50, we shall find that, just as arithmetical division is really the solution of an equation (§ 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 (§ 42). When, for instance, we find that the quotient, when 6+5x+7x2+13x3+5x4 is divided by 2+3x+x2, is made up of three terms +3, −2x, and +5x2, we are really obtaining successively the values of c0, c1, and c2 which satisfy the identity 6+5x+7x2+13x3+5x4＝(c0+c1x+c2x2)(2+3x+x2); and we could equally obtain the result by expanding the right-hand side of this identity and equating coefficients in the first three terms, the coefficients in the remaining terms being then compared to see that there is no remainder. 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. In this sense, the laws mentioned in § 54 apply also to algebraical division.
56. Extensions of the Binomial Theorem.—It has been mentioned in § 41 (ix.) that the binomial theorem can be used for obtaining an approximate value for a power of a number; the most important terms only being taken into account. There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly. The argument involves the theorem that, if θ is a positive quantity less than 1, θt can be made as small as we please by taking t large enough; this follows from the fact that t log θ can be made as large (numerically) as we please.
(i.) By algebraical division,
| 11+x＝1+0.x+0.x2+...+0.xr+11+x＝1 − x + x2−...+(−)rxr+(−)r+1xr+11+x | (24). |
If, therefore, we take 1/(1+x) as equal to 1−x+x2−...+(−)rxr, there is an error who whose numerical magnitude is |xr+1/(1+x)|; and, if |x| < 1, this can be made as small as we please.
This is the foundation of the use of recurring decimals; thus we can replace 411{＝3699＝36100/(1−1100)} by ·363636 (＝36/102+36/104+36/106), with an error (in defect) of only 36/(106 .99).
(ii.) Repeated divisions of (24) by 1+x, r being replaced by r+1 before each division, will give
(1+x)−2＝1−2x+3x2−4x3+...+(−)r(r+1)xr+(−)r+1xr+1{(r+1)(1+x)−1+(1+x)−2},
(1+x)−3＝1−3x+6x2−10x3+...+(−)r.12(r+1)(r+2)xr+(−)(r+1)xr+112(r+1)(r+2)(1+x)−1+(r+1)(1+x)−2+(1+x)−3}, &c.
Comparison with the table of binomial coefficients in § 43 suggests that, if m is any positive integer,
| (1+x)−m＝Sr+Rr | (25), |
| Rr ≡ (−)r+1xr+1{m[r](1+x)−1+(m−1)[r](1+x)−2+...+1[r](1+x)−m } | (27). |
This can be verified by induction. The same result would (§ 55) be obtained if we divided 1+0.x+0.x2+... at once by the expansion of (1+x)m.
(iii.) From (21) of § 43 (iv.) we see that |Rr| is less than m[r+1]xr+1 if x is positive, or than |m[r+1]xm+1(1+x)−m| if x is negative; and it can hence be shown that, if |x| < 1, |Rr| can be
