8. 2x2+3x-2, 4x3 16 x2 - 19 x + 5. 9. x3-3x2+4, 3x3-18 x2+36 x 24. 10. (a+b−c) x2+(ab-acbc) x + abc, 11. 23+x2-5x+3, 2x2+7x2-9. 12. 3x3-8x2-36x+5, 9 x3-50 x2+27 x - 10. 16. 7x+34x3y - 102 xy2 + 21 y3, 4x3 +33x2y + 29 xy2 — 42 y3. 17. aa2-5a+2, 3 a3- a2 - 8a+ 12. 18. 6x+4x2-51x+21, 2123-94 x2+144x-91. 19. 14x3-41 x2y+17 xy2-5 y3, 10 x3-31 x2y+23 xy2—20 y3. 20. 62-22+16, 10-19 x2+26 x 21. 2+2x2+2x+1, x3-4x24 x 8. 5. 22. 2 a3x3 + a2x2- ax+10, 3 a31⁄23 +11 a2x2 + 3 ax – 14. 23. a3x3-5a2x2-5 ax-6, 2 a3x3-3 a2x2-3 ax-5. 24. 30 x3- 25 ax2+8 a2x - a3, 18 x3-24 ax2 + 15 a2x — 3 a3. 25. 5 x3-31 x2+7x-6, 15x13x2+5x-2. 26. 36 a +9 a3- 27 a 18 a, 27 a3b2-9 a3b2 - 18 ab2. a1 - — 27. 3-1023 +15x+8, 25-2x-6x+4x2+13x+6. 28. 2 a3x* -2 a2bx3y + 2 ab2x2y2 - 2 b3xy3, 4 a2b2x23y2 - 2 ab3x2y3 — 2 b*xy*. 29. 223-9x-25x+42, 4x3-29 x2+37 x - 42. 30. 9 x9x-4x+4, 15 x3-19 x2+4. 31. 3x3-2x2-36x-35, 2x3-17 x2+23 x + 55. 32. 2+4x-4x+5, x+4x3-5 x2+3x+15. 33. 2x3-3x2-8x-3, 2x-9x+13x2-23 x 16. 34. 2 a3x3-7 a2x2+11 ax-15, 2 ax-7 a3x3+8 a2x2-12 ax-9. 35. 2+28 x2-8, x-2x+x2-2x. 36. 75 +140 x3-223x2+92 x − 12, 452-93 +65 x2-19x+2. 37. x2+4x+3, x2-4x-5, 2x2 - 5 x −7. 38. 2x2+5x-12, 2x-13x+15, 10 x2-23x+12. 39. 2-4x+3, 2x2+x2-7x+4, x3-2x2+1. 40. 2+5x-4x-10, 2x+5x2+2x+5, 2x+7x2+7x+5. 41. 23+x2-5x+3, x3-7x+6, x3-x2-10x+10. 42. 23-2x-34x+5, x+5x2+x+5, 3x+16x+6x+5. 43. 3x-14x3-9x+2, 2x-9 x3+x-19x+4, 5 x 23 x – 5 +2. 44. 2x+6x+4x2, 3x2+9x2+9x+6, 3x2+8x2+5x+2. 45. 10 a3+10 a3b2+20 a1b, 2 a3+2 b3, 4 b*+12 a2b2+4a3b+12 ab3. 46. x3-3x2-4x+12, x3-7 x2+16x-12, 2 x3-9 x2+7x+6. 47. a3x3-a2x2-11 ax-10, a3x3-5 a2x2+ax+10, — 2 a 3-5 a2x2-13 ax-5. 48. 2 x2+3x2+x+4, 2x-3x3-2x2+9x-12, 4x16x+25 x2 - 23 x +4. 13. The words Highest Common Factor in Algebra refer to the degree of the common factor. Thus, the H. C. F. of and is evidently 23-2x2-x+2=(x2-1)(x-2), 23-4x2-x+4= (x2-1)(x-4) - 1. That factor is of higher degree in x than any other common factor. The words Greatest Common Measure refer to the greatest numerical common measure when particular numerical values are substituted for the letters. Thus, if we substitute 6 for x in the expressions given above, we have 2-3 — 2x2-x+2= (x2-1) (x-2)= 35 x 4 = 140, and 3-4 — x + 4 = (x2 - 1)(x-4)= 35 x 2 = 70. = × The arithmetical G. C. M. of 70 and 140 is evidently 70. Now notice that when x = 6, the G. C. M. of the expressions is not the same in numerical value as the H. C. F.; for, when have an algebraic common factor, their numerical values for particular values of x may have a common numerical factor. Thus, when x = 6, x -2 and x 4 have the values 4 and 2 respectively, and therefore have the common factor 2. The words Greatest Common Measure should not therefore be used in the same sense as the words Highest Common Factor. 14. The following principles will be of use in subsequent work: (i.) In the process for finding the arithmetical G. C. M. of two integers, M and N, the remainder at any stage of the work can be expressed in the form ±(mM − nN), wherein m and n are positive integers, and the upper sign goes with the first, third, etc., remainders, and the lower sign with the second, fourth, etc. Let M be greater than N. Then, in the process for finding the G. C. M., let Q1 be the quotient and R1 the remainder of the first division, Q2 and R2 the quotient and the remainder, respectively, of the second division, and so on. It is to be kept in mind that the Q's and the R's are positive integers. Then, by Ch. III., § 4, Art. 13, we have In like manner, the value of each succeeding remainder, in terms of M and N, can be derived. In (4), m = 1, n = Q1; in (5), m = Q2, n = Q1 Q2 + 1, and so on. Since the Q's are positive integers, these values of m and n must be positive integers. Hence the truth of the principle enunciated. (ii.) If M and N be two positive integers, prime to each other, then two positive integers, m and n, can be found, such that mMnN=1. Since M and N are prime to each other, 1 is their G. C. M. Therefore, the next to the last remainder will be 1 (the last being 0). Consequently, by (i.) two positive integers, m and n, can be found, such that (iii.) If M and N be two positive integers, prime to each other, then any common factor of M and NR must be a factor of R. Therefore, m MR - nNR = ± R, or mR. M -N. NR=R. Since, by Art. 6 (ii.), any common factor of M and NR is a factor of mR.M-n. NR, the last equation shows that this factor is a factor of R. The following principles follow directly from (iii.): (iv.) If M be a factor of NR and be prime to N, it is a factor of R. (v.) If M be prime to R, S, etc., it is prime to RS .... (vi.) If each of the integers M, N, P be prime to each of the integers R, S, T, then MNP is prime to RST. (vii.) If M be prime to N, then MP is prime to NP, wherein p is a positive integer. § 3. LOWEST COMMON MULTIPLES. 1. The Lowest Common Multiple of two or more integral algebraic expressions is the integral expression of lowest degree which is exactly divisible by each of them. E.g., the lowest common multiple of ax, bx, and cx is evidently abcx. The words Lowest Common Multiple are frequently abbreviated to L. C. M. L. C. M. by Factoring. 2. The following examples will illustrate the method of finding the L. C. M. of two or more expressions which can be readily factored. Ex. 1. Find the L. C. M. of a3b, a2bc2, and ab2c1. The expression of lowest degree which is exactly divisible by each of the given expressions evidently cannot contain a lower power of a than a3, a lower power of b than b2, and a lower power of c than c1. Therefore, the required L. C. M. is a3b2c1. Observe that the power of each letter in the L. C. M. is the highest power to which it occurs in any of the given expressions. If the expressions contain numerical factors, the L. C. M. of these factors should be found as in Arithmetic. Ex. 2. Find the L. C. M. of 3 ab2, 6b (x + y), and 4 a2b (x − y) (x + y). The L. C. M. of the numerical coefficients is 12. The highest power of a in any of the expressions is a2; of b is b2; of x+y is (x + y); and of xy is xy. Consequently the required L. C. M. is 12 ab2 (x + y)2(x − y). In general, the L. C. M. of two or more expressions is obtained by multiplying the L. C. M. of their numerical coefficients by the product of all the different prime factors of the expressions, each to the highest power to which it occurs in any of them. EXERCISES XVIII. Find the L. C. M. of the following expressions: 1. 2a, 3b. 3. 4 ab, 2 a2b2, 12 a3b. 5. 7 ab, 3 a3bx, 2 ab, 2 a2x2. 7. 12 a3b2x, 18 x3a2b, 36 ab3x. 2. 4a3b, 2 ab2, 3 ax. 4. 14 a3, 21 a2, 5b, 7 a. 6. 7 a3m2, 21 x2m3, 343 xm. 8. 20 m12, 12 m3, 10 m2. 10. 90 xy, 50 x3m1, 6 x2y2m2. |