149. The principle, that the exact divisor reached by the process given in the rule is the highest common divisor, may be proved as follows: Let A and B represent any two polynomials freed of monomial factors, the degree of B being not higher than that of A. Divide A by B, and let the quotient be m and the remainder D; divide B by D. and let the quotient be n and the remainder E; divide D by E, and let the quotient ber and the remainder zero; that is, let E be an exact divisor of D. It is to be proved that E is the highest common divisor of A and B. PROCESS B) A(m mB D) B(n nD E)D(r Since the minuend is equal to the subtrahend plus the remainder, AmB+ D, and AmB = D ; Since the division has terminated, E is a common divisor of D and nD (Prin. 1); also of D and nD+E, or B (Prin. 2); also of B and mB (Prin. 1); also of B and mB + D, or A (Prin. 2). That is, E is a common divisor of B and A. Every common divisor of A and B is a divisor of mB (Prin. 1); and of A – mB, or D (Prin. 2). Therefore, every common divisor of A and B is a divisor of nD (Prin. 1); and of B – nD, or E (Prin. 2). But, since no divisor of E can be of higher degree than E itself, E is the highest common divisor of A and B. 150. The principle, that the highest common divisor of several expressions may be obtained by finding the highest common divisor of two of them, then of this result and a third expression, and so on, may be proved as follows: Let P be the highest common divisor of A and B, and Q the highest common divisor of P and a third expression C. Then, since P contains all the common factors of A and B, and Q contains of these particular factors only such as are factors of C also, Q is the highest common divisor of A, B, and C. This method may be extended to embrace any number of expressions. Find the H. C. D. of 12. 2x3-72+2x+3 and 2+7x2-5x-4. 13. 9+18x2-x-10 and 3x+13x2+2x-8. 14. 1 2x 5 x2+6x3 and 1+5x+2x2 – 8 x3. 15. 1 − 4 x + x2 + 6 x3 and 1+3x-6 x2 - 8 x3. 16. 1 x 14 x2+24x3 and 363-24 x2 + x + 1. 17. m3-4 m2 - 20 m +48 and m3- m2 - 14 m + 24. 18. 3a+20 a2-a-2 and 3a+17 a2+21a-9. 19. 8 ax2+22 ax + 15 a and 6 ba2+11 bx +3 b. 20. 20 b c 2 bc-4 c and 8 a2bc-4 a bea2c. 21. 21 ax-17 ax2 - 5 ax3+ax and 7 ax + 34 ax2 - 5 αx3. 22. 23-7x+6, x2 - 2 x3-9 x2+18x, a3 + x2 - 4 x − 4. 23. 2-5x+4, 24-2x2+1, +4x3-3x-2.. 24. 1+4x2+5a3, 2+5x+3x, a -4x+5a2-2. 25. 3x-8x2+4x3, 3-8 x-8a2+8x3, 16 x 48 x3 +81. 26. 23 - 6x2 5–14, − 10 +20x+, 310x 231. 27. Find the H. C. D. of x+xo1+x3-x−2 and 2x+x-x3 — x2 — 1. 2x3 x2 + x + 2 and a +3 x1 +3 x3 + x2 − x − 1. 29. xx x2 - 7 x 4 and 2+ 3 x1 + 3 x3 + 3 x2 - 7 x − 4. 30. -2x-2x3-11x2-x-15 and 2-7x+4x3-15x2+x-10. 31. a-3a-3 a3-3a2-19a-15 and a+3a-3 a3+9a2-a-15. 32. 5a+a+-11 a3+9 a2-8 a+4 and 2a-a1-5a3+8 a2-4 a. 5x + 4 and x-23-3x2-5x-12. 33. 34. a3 +3a2 - 2 a 6 and a +4 a* + 4 a3 + 4 a2 — a 35. 1-4 a3 + 3 at and 1 + a — a2 − 5 a3 + 4 a1. 36. 2a+3a2+5a3-a and 4-4a+a2 — 9 a*. – 5 – 7 LOWEST COMMON MULTIPLE 151. 1. What number exactly contains 2, 5, a, and b, or is a multiple of 2, 5, a, and b? 2. What different prime factors must enter into every number that will contain 4 a3b, a2b2, and 10 ab3, or must be found in every common multiple of 4 a3b, a2b2, and 10 ab3? 3. What is the lowest power of a that common multiples of 4 a3b, a2b2, and 10 ab3 can contain? What is the lowest power of b? of 2? of 5? What, then, is the lowest common multiple of 4 a3b, a2b2, and 10 ab3? To what is the lowest common multiple of two or more expressions equal? 152. An expression that exactly contains each of two or more given expressions is called a Common Multiple of them. 6 abx is a common multiple of a, 3b, 2x, and 6 abx. These numbers may have other common multiples, as 12 abx, 6 a2b2x, 18 a3bx2, etc. 153. The expression of lowest degree that will exactly contain each of two or more given expressions is called their Lowest Common Multiple. 6 abx is the lowest common multiple of a, 3b, 2x, and 6 abx. The abbreviation L. C. M. is used for Lowest Common Multiple. The lowest common multiple in algebra corresponds to the least common multiple in arithmetic. But, since letters may represent any numbers, as, for instance, numbers not prime to each other or fractions, the term least is not applicable to algebraic common multiples. = Thus, the algebraic lowest common multiple of a2b2, ab3, and bx is a2b3x. If a = 4, b 3, and x = 2, a2b3x, the lowest common multiple of the given expressions, is equal to 864. If, however, the values of a, b, and x are substituted for those letters, the given expressions become 144, 108, and 6; and their least common multiple is 432. It is thus seen that the lowest common multiple of two or more expressions is not necessarily their least common multiple. 154. PRINCIPLE. The lowest common multiple of two or more algebraic expressions is the product of all their different prime factors, using each factor the greatest number of times it occurs in any of the expressions. 155. To find the lowest common multiple of expressions that may be factored readily by inspection. EXAMPLES 1. What is the L. C. M. of 12 xyz, 6 a2xy2, and 8 axyz2? L. C. M. = 2.2 · 2 · 3 · a2 · x2 · y2 • za = 24 a2x2y2za EXPLANATION. The lowest common multiple of the numerical coefficients is found as in arithmetic. It is 24. The literal factors of the lowest common multiple are each letter with the highest exponent it has in any of the given expressions (Prin.). They are, therefore, a2, x2, y2, and . The product of the numerical and literal factors, 24 a2x2y2z4, is the lowest common multiple of the given expressions. 2. What is the L. C. M. of a2 - 2 xy + y2, x2 — y2, and x3 + y3? RULE. Factor the expressions as far as may be necessary to discover their different prime factors. Find the product of all their different prime factors, using each factor the greatest number of times it occurs in any of the given expressions. The factors of the L. C. M. may often be selected without separating the expressions into their prime factors. Find the L. C. M. of 3. axy, a2xy, and axy. 4. 10 a b c2, 5 ab'c, and 25 b3c3d3. 9. ay and x2-2xy + y2. n+ 10. x2-y2, x2+2xy + y2, and x2-2xy + y2. 11. a2 n2 and 3 a3 + 6 a2n + 3 an2. 12. x4 1 and a2x2+ a2 — b2x2 — b2. 13. a2+1, ab — b, a2 + a, and a2 - 1. - 14. 2x+y, 2 xy-y2, and 4 x2 — y2. |