Thus we find that the greatest common divisor is x-3. Had we suppressed +19 instead of -19, in the final remainder of the second operation, we should have obtained-x+3, or 3-x for the greatest common divisor. It should be remembered, however, that the term greatest, in this connection, has reference to exponents and coefficients, and not to the algebraic value; (98). Consequently either x-3, or 3-x may be considered the greatest common divisor of the given polynomials. And it is immaterial what sign is given to any monomial factor which we may suppress or introduce at any stage of the work. 105. From these principles and illustrations we deduce the following general RULE. I. Arrange the two polynomials with reference to the same letter; then suppress all the monomial factors of each, and if any factor suppressed is common to the two polynomials, set it aside as one factor of the common divisor sought. II. Divide the greater of the resulting polynomials by the less, and continue the division till the first term of the remainder is of a lower degree than the first term of the divisor; observing to suppress the monomial factors in every remainder, and to introduce into any dividend, if necessary, such a factor as will render its first term exactly divisible by the first term of the divisor. III. Take the final remainder in the first operation as a new divisor, and the former divisor as a new dividend, and proceed as before; and thus continue till the division is exact. The last divisor, multiplied by the common factor, if any, set aside at the beginning, will be the greatest common divisor required. IV. If more than two polynomials are given, find the greatest common divisor of the first and second, and then the greatest common divisor of this result and the third polynomial, and so on. The last will be the greatest common divisor required. EXAMPLES FOR PRACTICE. Find the greatest common divisor, 1. Of x-2x-4x+11x-6 and x-8x+17x-10. Ans. x-3x+2. 2. Of 6x+x3-44x+21 and 6x3-26x2+46x-42. Ans. 3x-7. 3. Of x-6ax+10a'x-3a and 3ac-14a'x+15a'. Ans. x-3a. 4. Of x-8x+14c+16x-40 and x-8x+19x-14. 5. Of a3+5a3+5a+1 and a3+1. 6. Of 2a*—5a3b—3a*b2+7ab*+36′ and 4a3—2a3b—4ab’—3b'. Ans. a+1. Ans. 2a-3b. 7. Of 3x-4x'y+3xy-2y' and 4x-7xy+3y'. Ans. x-y. 8. Of 4x*—2x*+4x3-27x2+4x-7 and 2x+6x'—19x3+4x -5. Ans. 2x-4x+x-1. 9. Of a'c-4a3cm+3acm3 and a‘c'—6a2c3m+5c3m2. Ans. c(a'-m). 10. Of x*—4x3-16x2+7x+24 and 2x3-15x+9x+40. Ans. x-5x-8. 11. Of 15x+71x+60x2-56 and 3x-17x-20x+84. Ans. 3x2+7. 12. Of 3a+14a2m2-5m*, 6a-14a'm'+4m*, and 3a2-22a'm' +7m*. Ans. 3a2-m2. 13. Of 2a'x3-2a3bx3y+ab3xy3—b'y' and a'bx ̈y-2ab3xy2+b3y". Ans. ux-by. 14. Of 9a+12a+10a3+4a+1 and 3a +8a8+14a2+8a+3. Ans. 3a+2a+1. LEAST COMMON MULTIPLE. 106. A Multiple of any quantity is another quantity exactly divisible by the given quantity. It follows from this definition that if one quantity is a multiple of another, the multiple must be equal to the product of the other quantity by some entire factor, Thus, if A is a multiple of B, then A=Bm, in which m is entire. 107. A Common Multiple of two or more quantities is one which is exactly divisible by each of them. 108. The Least Common Multiple of two or more quantities is the least quantity which is exactly divisible by each of them. CASE I. 109. When the quantities can be factored by inspection. From the principles of exact division, we may make the following inferences: 1. A multiple of any quantity must contain all the factors of that quantity. 2.-A common multiple of two or more quantities must contain all the factors of each of the quantities. 3.--The least common multiple of two or more quantities must contain all the factors of each of the quantities, and no other factors. Hence the following RULE. I. Find by inspection all the different prime factors that enter into the given quantities, and affect each with an exponent equal to the greatest which it has in any of the quantities. II. Multiply together the factors thus obtained, and the product will be the least common multiple required. EXAMPLES FOR PRACTICE. 1. What is the least common multiple of a'+ab, a3d—b3d, and a2c-2abc+b*c? Factoring, we have a2 +ab =a(a+b) =d(a—b)(a+b) ac-2abcb'c=c(a-b)' The highest powers of the different prime factors are a, d, c, (a—b)3, and (a+b); and we have acd(a—b)' (a+b)=a*cd—a2bcd—a2bcd+ab'cd, Ans. 2. Find the least common multiple of 2a*bc, 5a3c3, 10ab3d, and 15abcd. Ans. 30a b'c'd. 3. Find the least common multiple of 3x3y, 15xy3, 10xyz3, and 5x y3z. Ans. 30x3y'z'. 4. Find the least common multiple of x+xy, xy-y3, and x3-y2. Ans. x'y-xy'. 5. Find the least common multiple of x-a1, x2—a3, x2+a3, and x- -2a2x2+a*. Ans. x-a2x-aˆx2+a". 6. Find the least common multiple of x3-x, x3 —-1, and ∞3+1. 7. Find the least common multiple of x+2x+1, x*—2x2+1, x2+2x+1, x2-2x+1, x+1, and x-1. Ans. x-2x+1. 8. What is the least common multiple of 4x+2x, 6x3-4x, and 6x2+4x? Ans. 36x+2x3-8x. 9. What is the least common multiple of x2—4a2, (x+2a)3, and (x-2a)? 10. What is the least common multiple of a*—b′, a3—b3, a3—l3, . and a-b? Ans. ao+ab+a*b3—a*b*—ab®—b".` CASE II. 110. When the quantities can not be factored by inspection. The rule for this case may be deduced as follows: 1.--If two polynomials are prime to each other, their product must be their least common multiple. 2. If two polynomials have a common divisor, their product must contain the second power of this common divisor; their least common multiple will therefore be obtained, by suppressing the first power of the common divisor in the product, or in one of the given quantities before multiplication. 3.—If we find the least common multiple of two polynomials, and then the least common multiple of this result and a third polynomial, and so on, the last result will evidently contain all the factors of the given polynomials, and no other factors. It will, therefore, be the least common multiple of the polynomials (109, 3). Hence the following RULE. I. When only two polynomials are given : Find the greatest common divisor of the given polynomials; suppress this divisor in one of the polynomials, and multiply the result by the other polynomial. II. When three or more polynomials are given : Find the least common multiple of any two of the polynomials; then find the least common multiple of this result and a third polynomial; and so on, till all the polynomials have been used. The last result will be the least common multiple required. NOTE.-It will generally be found preferable to commence with the greatest and next greatest of the given quantities. EXAMPLES FOR PRACTICE. Find the least common multiple 1. Of x3+x2-4x+6 and x-5x+8x-6. Ans. x-2x-7x2+18x-18. 2. Of x-2x-19x+20 and x2-12x+35. Ans. x-9x-5x+153x-140. 3. Of 6am-am-1 and 2a'm'+3am2-2. Ans. 6am +11am-3am'-2. 4. Of 2x3-5x2-x+1 and x3-5x2+7x-2. Ans. 2x-9x+9x2+3x-2. 5. Of 3x+6x-5x-10 and 6x-4x2-10. Ans. 6x+12xa—4x3—8.x2—10x—20. 6. Of x+7x+10, x2-2x-8, and x2+x-20. Ans. x+3x-18x-40. 7. Of a2-3ab+2b', a2-ab-262, and a2-b'. Ans. a-2ab-a b2+2b3. 8. Of 2x2-7xy+3y2, 2x2-5xy+2y3, and x3-5xy+6y3. Ans. 2x3-11x3y+17xy3—6y3. |