and the solution of the equation is the determination of the points where the ordinates of the graph are zero. The second class of cases comprises equations involving two unknowns; here we have to deal with two graphs, and the solution of the equation is the determination of their common ordinates.
Graphic methods also enter into the consideration of irrational numbers (§ 65).
III. Elementary Algebra of Positive Numbers
36. Monomials.—(i.) An expression such as a.2.a.a.b.c.3.a.a.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) which are multiplied together being its factors. 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.
(ii.) By means of the commutative law we can collect like terms of a monomial, numbers being regarded as like terms. Thus the above expression is equal to 6a5bc2, which is, of course, equal to other expressions, such as 6ba5c2. The numerical factor 6 is called the coefficient of a5bc2 (§ 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors.
(iii.) The multiplication and division of monomials is effected by means of the law of indices. Thus 6a5bc2 ÷ 5a2bc = 65a3c, since b0 = 1. It must, of course, be remembered (§ 23) that this is a statement of arithmetical equality; we call the statement an “identity,” but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result.
In order that a monomial containing am as a factor may be divisible by a monomial containing ap as a factor, it is necessary that p should be not greater than m.
(iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple. We disregard numerical coefficients, so that by the H.C.F. or L.C.M. of 6a5bc2 and 12a4b2cd we mean the H.C.F. or L.C.M. of a5bc2 and a4b2cd. The H.C.F. is then an expression of the form apbqcrds, where p, q, r, s have the greatest possible values consistent with the condition that each of the given expressions shall be divisible by apbqcrds. Similarly the L.C.M. is of the form apbqcrds, where p, q, r, s have the least possible values consistent with the condition that apbqcrds shall be divisible by each of the given expressions. In the particular case it is clear that the H.C.F. is a4bc and the L.C.M. is a5b2cd.
The extension to multinomials forms part of the theory of factors (§ 51).
37. Products of Multinomials.—(i.) Special arithmetical results may often be used to lead up to algebraical formulae. Thus a comparison of numbers occurring in a table of squares
12＝1 |
112＝121 |
suggests the formula (A + a)2＝A2 + 2Aa + a2. Similarly the equalities
99 × 101＝9999＝10000 − 1
98 × 102＝9996＝10000 − 4
97 × 103＝9991＝10000 − 9
. . .
. . .
. . .
lead up to (A−a) (A + a)＝A2−a2. These, with (A−a)2 = A2−2Aa + a2, are the most important in elementary work.
(ii.) These algebraical formulae involve not only the distributive law and the law of signs, but also the commutative law. Thus (A + a)2＝(A + a)(A + a)＝A(A + a) + a(A + a)＝AA + Aa + aA + aa; and the grouping of the second and third terms as 2Aa involves treating Aa and aA as identical. This is important when we come to the binomial theorem (§ 41, and cf. § 54 (i.)).
(iii.) By writing (A+a)2＝A2 + 2Aa + a2 in the form (A+a)2＝A2 + (2A+a)a, we obtain the rule for extracting the square root in arithmetic.
(iv.) When the terms of a multinomial contain various powers of x, and we are specially concerned with x, the terms are usually arranged in descending (or ascending) order of the indices; terms which contain the same power being grouped so as to give a single coefficient. Thus 2bx − 4x2 + 6ab + 3ax would be written −4x2 + (3a+2b)x + 6ab. It is not necessary to regard −4 here as a negative number; all that is meant is that 4x2 has to be subtracted.
(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. If any power is absent, we treat it as present, but with coefficient 0. Thus, to multiply x3 − 2x + 1 by 2x2+4, we write the process
- giving 2x5 + 2x2 − 8x + 4 as the result.
38. Construction and Transformation of Equations.—(i.) The statement of problems in equational form should precede the solution of equations.
(ii.) The solution of equations is effected by transformation, which may be either arithmetical or algebraical. The principles of arithmetical transformation follow from those stated in §§ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities. The principle of algebraic transformation has been stated in § 22; it is that, if A＝B is an equation (i.e. if either or both of the expressions A and B involves x, and A is arithmetically equal to B for the particular value of x which we require), and if B＝is an identity (i.e. if B and C are expressions involving x which are different in form but are arithmetically equal for all values of x), then the statement A=C is an equation which is true for the same value of x for which A=B is true.
(iii.) A special rule of transformation is that any expression may be transposed from one side of an equation to the other, provided its sign is changed. This is the rule of transposition. Suppose, for instance, that P+Q−R+S＝T. This may be written (P+Q−R)+S＝T; and this statement, by definition of the sign −, is the same as the statement that (P+Q−R)＝T−S. Similarly the statements P+Q−R−S＝T and P+Q−R＝T+S are the same. These transpositions are purely arithmetical. To transpose a term which is not the last term on either side we must first use the commutative law, which involves an algebraical transformation. Thus from the equation P+Q−R+S＝T and the identity P+Q−R+S＝P−R+S+Q we have the equation P−R+S+Q＝T, which is the same statement as P−R+S＝T−Q.
(iv.) The procedure is sometimes stated differently, the transposition being regarded as a corollary from a general theorem that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation. The objection to this (cf. § 21 (ii.)) is that we do not need the general theorem, and that it is unwise to cultivate the habit of laying down a general law as a justification for an isolated action.
(v.) An alternative method of obtaining the rule of transposition is to change the zero from which we measure. Thus from P+Q−R+S＝T we deduce P+(Q−R+S)＝P+(T−P). If instead of measuring from zero we measure from P, we find Q−R+S＝T−P. The difference between this and (iii.) is that we transpose the first term instead of the last; the two methods corresponding to the two cases under (i.) of § 15 (2).
(vi.) In the same way, we do not lay down a general rule
