Greatest Common Divisor. The third line of col. 1 is the remainder of the division of the 1st line of col. 1 by the 1st line of col. 2; and this remainder, reduced by the suppression of the factor x is the 4th line of col. 1. The 5th line of col. 2 is the remainder of the division of the 1st line of col. 2 by the 4th line of col. 1, and this remainder, reduced by the suppression of the factors x+1 is the last line of col. 2. The 4th line of col. 1 is exactly divisible by the last line of col. 2, and therefore the greatest common divisor is the product of (x2+3x) y by y + 2. Ans. (x2+3x) (y2+2y). 2. Find the greatest common divisor of the polynomials a2+b2 + c2+2ab+2ac+2 b c and a2-b2 — c2 — Ans. a+b+c. 2 b c. 3. Find the greatest common divisor of the polynomials a1 — 2 b2 | a2 + 64 and a33ba2+3 b2 | a+ 63 67. Problem. To reduce fractions to a common denominator. Solution. Multiply both terms of each fraction by the product of all the other denominators. Common Denominator. For the value of each fraction is, from art. 55, not changed by this process; and as each of the denominators thus obtained is the product of all the denominators, the fractions are all reduced to the same denominator. 68. But fractions can be reduced to a common denominator which is smaller than their continued product, whenever their denominators have a common multiple less than this product. For, by art. 55, Fractions may be reduced to a common denominator, which is a common multiple of their denominators, by multiplying both their terms by the quotients, respectively obtained from the division of the common denominator by their denominators. 69. Corollary. An entire quantity may, by the preceding article, be reduced to an equivalent fractional expression having any required denominator, by regarding it as a fraction, the denominator of which is unity. 71. Problem. To find the least common multiple of given quantities. Solution. When the given quantities are decomposed into their simplest factors, as is the case with monomials, their least common multiple is readily obtained; for it is obviously equal to the product of all the unlike factors, each factor being raised to a power equal to the highest power which it has in either of the given quantities. But the common factors can always be obtained from the process of finding the greatest common divisor. 1. Find the least common multiple of 2 a3 b2 c x, 3 a5 b c3 x2, 6 a c x = 2.3 a c x, 9 c7 x1o — 32 c7 x1o, 24 a3 = 23.3 a8. Ans. 23.32. a8 b2 c7 x10 = 72 a8 b2 c7 x 10. 2. Find the least common multiple of 16 a x, 40 b5 x, 25 a b3 x2. Ans. 400 a7 b5 x2. xn 3. Find the least common multiple of ", -1, xn−2, xn-3, x. Ans. x. 4. Find the least common multiple of 6 (a + b) xm, 54 (a-b)3, (a+b)7, 81 (a—b)3 xm+2, 8(a+b)5 xm8 Ans. 648 (a+b)7 ( a − b )3 xm+2. Sum and Difference of Fractions. 5. Find the least common multiple of a2+2 a b + b2, a2 + 4 a b + 4 b2, a2 — b2, a2 + 3 a b + 2 b2, a3 + a2 b 73. Problem. To find the sum or difference of given fractions. Solution. When the given fractions have the same denominator, their sum or difference is a fraction which has for its denominator the given common denominator, and for its numerator the sum or the difference of the given numerators. When the given fractions have different denominators, they are to be reduced to a common denominator by arts. 67 and 68. |