the given expressions and also the G. C. M. of the other two; then the G. C. M. of the two expressions thus found will be the G. c. M. of the four given expressions.

121.

The definition and operations of the preceding articles of this chapter relate to polynomial expressions. The meaning of the term greatest common measure in the case of simple expressions will be seen from the following example:

Required the G. C. M. of 432a b'xy, 270a b'x'z and 90a3bx3.

We find by Arithmetic the G. C. M. of the numerical coefficients 432, 270, and 90; it is 18. After this number we write every letter which is common to the simple expressions, and we give to each letter respectively the least index which it has in the simple expressions. Thus we obtain 18a bx, which will divide all the given simple expressions, and is called their greatest com

mon measure.

EXAMPLES OF THE GREATEST COMMON MEASURE.

Find the G. C. M. in the following examples:

1. Of x2 - 3x + 2 and x2-x-2.

2.

... x3 + 3x2 + 4x + 12 and x3 + 4x2 + 4x + 3.

7.

3x3 13x2+23x-21 and 6x3 + x2 - 44x + 21.

8. ... x2 - 3x3 + 2x2 + x − 1 and x3- x2 – 2x + 2.

9. ... x2-7x3 + 8x2 + 28x − 48 and x3- 8x2 + 19x – 14.

10. Of x-x3 + 2x2 + x + 3 and x2+ 2x3

11.

...

-x-2.

-x+2.

4x+9x3 + 2x2 - 2x-4 and 3x3 +5x2

2x-12x+19x2-6x+9 and 4x3- 18x2 + 19x - 3.

6x+x3 -x and 4x3- 6x2 - 4x + 3.

x2+ ax3-axy-y3 and x+2x3y - a2x2+x3y3-2axy-y*.

2x5-11x2-9 and 4x+111+81.

2a+3a3x- 9a2x2 and 6ax - 17a3x + 14a2x2 - 3αx*.

2×3 + (2α − 9) x2 — (9a+6) x + 27 and 2x2 – 13x + 18.

18. ... a3×3— a3bx3y + ab3xy3 — b3y3 and 2a3bx3y — ab3xy3 — b3y3.

12x2-15yx + 3y2 and 6x3 – 6yx2 + 2y3x — 2y3.

x5+ 3x - 8x2 - 9x-3 and x3- 2x2-6x3+4x2 + 13x+6.

6x-4x-11x3-3x2-3x-1 and 4x+2x3-18x2+3x-5.

x2 - ax3- a2x2 - a3x-2a and 3x3-7ax2 + 3a2x - 2a3.

VII. LEAST COMMON MULTIPLE.

122. In Arithmetic the least common multiple of two or more whole numbers is the least number which contains each of them exactly. The term is also used in Algebra, and its meaning in this subject will be understood from the following definition of the least common multiple of two or more Algebraical expressions. Let two or more Algebraical expressions be arranged according to descending powers of some common letter; then the expression of lowest dimensions in that letter which is divisible by each of these expressions is their least common multiple.

Т. А.

4

123. The letters L. C. M. will often be used for shortness instead of the term least common multiple; the term itself is not very appropriate for the reason already given in Art. 106.

Any expression which is divisible by another may be said to be a multiple of it.

124. We shall now shew how to find the L. C. M. of two Algebraical expressions. Let A and B denote the two expressions, and D their greatest common measure. Suppose AaD and B = bD. Then from the nature of the greatest common measure, a and b have no common factor, and therefore their least common multiple is ab. Hence the expression of lowest dimensions which is divisible by aD and bD is abD.

Hence we have the following rule for finding the L. C. M. of two Algebraical expressions: find their G. C. M.; divide either expression by this G. C. M., and multiply the quotient by the other expression. Or thus:-divide the product of the expressions by their G. C. M.

125. If M be the least common multiple of A and B, it is obvious that every multiple of M is a common multiple of A and B.

126. Every common multiple of two Algebraical expressions is a multiple of their least common multiple.

Let A and B denote the two expressions, M their L. C. M.; and let N denote any other common multiple. Suppose, if possible, that when N is divided by M there is a remainder R; let q denote the quotient. Thus N = qM + R; therefore R = N-qM. Now A and B measure M and N, and therefore (Art. 109), they measure R. But R is of lower dimensions than M; thus there is a common

multiple of A and B of lower dimensions than their L. C. M. This is absurd; hence there can be no remainder R; that is, N is a multiple of M.

127. Next suppose we require the L. C. M. of three Algebraical expressions A, B, C. Find the L. C. M. of two of them, say A and B; let M denote this L. C. M.; then the L. C. M. of M and C is the required L. C. M. of A, B and C.

For every common multiple of M and C is a common multiple of A, B and C, (Art. 125). And every common multiple of A and B is a multiple of M, (Art. 126); thus every common multiple of A, B and C is a common multiple of M and C. Therefore the L. C. M. of M and C is the L. C. M. of A, B and C.

128. By resolving Algebraical expressions into their component factors, we may sometimes facilitate the process of determining their G. C. M. or L. C. M. For example, required the L. C. M. of x2- a2 and x3 – a3.

Since

x2 — a2 = (x − a) (x + a) and x3 — a3 = (x − a) (x2 + ax + a3),

we infer that x

a is the G. C. M. of the two expressions; consequently their L. C. M. is (x + a) (x3 — a3), that is,

129. The preceding articles of this chapter relate to polynomial expressions. The meaning of the term least common multiple in the case of simple expressions will be seen from the following example. Required the L. C. M. of 432a b'xy, 270a b3x°z and 90a3bx3. We find by Arithmetic the L. C. M. of the numerical coefficients 432, 270 and 90; it is 2160. After this number we write every letter which occurs in the simple expressions, and 'we give to each letter respectively the greatest index which it has in the simple expressions. Thus we obtain 2160a*b3xyz, which is divisible by all the given simple expressions, and is called their least common multiple.

130. The theories of the greatest common measure and of the least common multiple are not necessary for the subsequent chapters of the present work, and any difficulties which the student may find in them may be postponed until he has read the theory of equations. The examples however attached to the preceding chapter and to the present chapter should be carefully worked on account of the exercise which they afford in all the fundamental processes of Algebra.

EXAMPLES OF THE LEAST COMMON MULTIPLE.

1. Find the L. C. M. of 6x3- x 1 and 2x2 + 3x - 2.

of

2x3 + (2α − 3b) x3 − (2b2 + 3ab) x + 363 and 2x2 - (36 - 2c) x-3bc.

13. Required the L. C. M. of

6 (a3 — b3) (a − b)3, 9 (a* — b1) (a − b)2 and 12 (a2 — b3)3.