21. X 2x2+3x+1 2x2-17x+21 4x2-20x+25° a2+ab-ac (a+c)2 − b2 ^ ab — b2 – bc ` a2+2ab+b2- c2 a2-2ac+c2- b2 a2-b2-c2-2bc b2-2bc+c2 - a2 * x2+2x+1 x2+2x+4 XC- - 5 (a - b)2 - c2 x2+24x+128 x2+12x-64 x2-16x+64 CHAPTER XX. LOWEST COMMON MULTIPLE. 159. IN Chap. XI. we defined the lowest common multiple of two or more algebraical expressions to be the expression of lowest dimensions which is divisible by each of them without remainder. We there shewed how in the case of simple expressions the lowest common multiple could be written down by inspection. The lowest common multiple of compound expressions which are given as the product of factors, or which can be easily resolved into factors, can be readily found by a similar method. Example 1. The lowest common multiple of 6x2 (a — x)2, 8a3 (a − x)3 and 12ax (a - x)5 is 24a3x2 (a — x)3. For it consists of the product of (1) the numerical L. C. M. of the coefficients; (2) the lowest power of each factor which is divisible by every power of that factor occurring in the given expressions. Example 2. Find the lowest common multiple of 3a2+9ab, 2a3-18ab2, a3+6a2b+9ab2. 9. x2+2x, x2+3x+2. x2+4x+4, x2+5x+6. 11. 10. x2-3x+2, x2 − 1. 12. x2-5x+4, x2-6x+8. 13. x2-x-6, x2+x−2, x2 - 4x+3. 21. 22. 12x2+3x-42, 12x3+30x2+12x, 32x2 - 40x - 28. 3x2+26x3+35x2, 6x2+38x-28, 27x3+27x2 - 30x. 23. 60x+5x3-5x2, 60x2y+32xy +4y, 40x3y - 2x2y-2xy. 24. 8x2-38xy+35y2, 4x2-xy-5y2, 2x2-5xy-7y2. 25. 12x2-23xy+10y2, 4x2-9xy+5y2, 3x2-5xy+2y2. 26. 6ax3+7a2x2 - 3a3x, 3a2x2+14a3x-5aa, 6x2+39ax+15a2. 27. 4ax2y2+11axy2-3ay2, 3x3y3 - 7x2y3 - 6xy3, 24ax2 - 22ax+4a. 160. When the given expressions are such that their factors cannot be determined by inspection, they must be resolved by finding the highest common factor. Example. Find the lowest common multiple of 2x2+x3- 20x2 - 7x + 24 and 2x4 + 3x3 - 13x2 – 7x+15. The highest common factor is x2 + 2x – 3. By division, we obtain 2x4+x3- 20x2 − 7x+ 24 = (x2+ 2x − 3) (2x2 - 3x-8). -- 2x4+3x3- 13x2 - 7x+15= (x2 + 2x − 3) (2x2 − x − - 5). Therefore the L. C. M. is (x2 + 2x − 3) (2x2 - 3x-8) (2x2 — x − 5). *161. We may now give the proof of the rule for finding the lowest common multiple of two compound algebraical expressions. Let A and B be the two expressions, and F their highest common factor. Also suppose that a and b are the respective quotients when A and B are divided by F; then A=aF, B=bF. Therefore, since a and b have no common factor, the lowest common multiple of A and B is abF, by inspection. *162. There is an important relation between the highest common factor and the lowest common multiple of two expressions which it is desirable to notice. Let Fbe the highest common factor, and X the lowest common multiple of A and B. Then, as in the preceding Article, Hence the product of two expressions is equal to the product of their highest common factor and lowest common multiple. hence the lowest common multiple of two expressions may be found by dividing their product by their highest common factor; or by dividing either of them by their highest common factor, and multiplying the quotient by the other. *163. The lowest common multiple of three expressions A, B, C may be obtained as follows. First, find X the L. C. M. of A and B. Next find Y the L.C.M. of X and C; then I will be the required L. C. M. of A, B, C. For Y is the expression of lowest dimensions which is divisible by X and C, and X is the expression of lowest dimensions divisible by A and B. Therefore Y is the expression of lowest dimensions divisible by all three. EXAMPLES XX. b. 1. Find the highest common factor and the lowest common multiple of 2-5x+6, x2-4, x3 – 3x – 2. 2. Find the lowest common multiple of ab (x2+1)+x (a2+b2) and ab (x2 − 1) + x (a2 — b2). 3. Find the lowest common multiple of xy - bx, xy-ay, y2 - 3by + 2b2, xy-2bx - ay+2ab, xy- bx - ay + ab. |