Now the remainder k is always less than m the divisor; hence, since a and b neasure m and m', it is evident by (2) that a and b measure m'--hm, or k; herefore k is a common multiple of a and b, and it has been proved to be less than m, the least common multiple, which is absurd; hence m must measure m', or m' is a multiple of m. 35. To find the least common multiple of three or more quantities. Let a, b, c, d, &c., be the proposed quantities; find m the least common multiple of a and b Then, since every multiple of a and b measures m, their least common multiple, the quantity sought, x, measures m; but x also measures c; therefore x measures both c and m, and thence it measures m'; but x measures d and m', and therefore must measure m"; hence x cannot be less than m", and therefore m" is the least common multiple. EXAMPLES. (1.) Find the least common multiple of 2a2, 4a3 b2, and 6a b3. Here taking 2a2 and 4a3 b2, we find d=2a2, and, therefore, Again, taking m, or 4a3 b2, and 6a b3, we find d=2a b2; hence cm 6a b3 × 4a3 b2 m2 = d 2ab2 = 12a3 b3 = answer required. (2.) Find the least common multiple of a—x, a2—x2, and a3—x3. Taking a―x and a2—x2, we have d=a—x; and hence Again, taking a2—x2 and a3—x3, we find d=a-x; hence (a3—x3) (a2—x2) = (a+x) (a3—x3)= answer sought. a-x (3.) Find the least common multiple of 15a2 b2, 12a b3, and 6a3 b. (4.) Find the least common multiple of 6a2 x2 (a−x), 8x3 (a2—x2) and 12 (a—x)2. (5.) Find the least common multiple of x3-x2y-xy2+y3, x3—x2y+xy2—y3; and x1—y1. (6.) Find the least common multiple of (a+b), (a2—b2), (a—b)2, and a'+3a2b+3a b2+b3. (3.) 60a3 b3. ANSWERS. (5.) x-xy-x1y+y3. (4.) 24a2 x' (a-x) (a2—x2) (6.) (a+b) (a2—b2)2. OF ALGEBRAIC FRACTIONS. 36. Algebraic fractions differ in no respect from arithmetical fractions; and all the observations which we have made upon the latter, apply equally to the former. We shall therefore merely repeat the rules already deduced, adding a few examples of the application of each. It may be proper to remind the reader, that all our operations with regard to fractions were founded upon the three following principles: 1. In order to multiply a fraction by any number, we must multiply the numerator, or divide the denominator of the fraction by that number. 2. In order to divide a fraction by any number, we must divide the numerator, or multiply the denominator of the fraction by that number. 3. The value of a fraction is not changed, if we multiply or divide both the numerator and denominator by the same number.* REDUCTION OF FRACTIONS. 1. To reduce a fraction to its lowest terms. 37. RULE.-Divide both numerator and denominator by their greatest common measure, and the result will be the fraction in its lowest terms. When the numerator and denominator are, one or both of them, monomials, their greatest common factor is immediately detected by inspection; thus, a2bc a2bc с in its lowest terms. 5a2b2a2bx5b5b So also, ax2 ххах ax in its lowest terms. ax+x2 ̄ ̄x(a+x) ̄ ̄α +x If, however, both numerator and denominator are polynomials, we must have recourse to the method of finding the greatest common measure of two algebraic quantities, developed in a former article. Thus, let it be required to reduce the following fraction to its lowest terms: 6a3-6a2y+2ay2-2y3 * These principles will be obvious from the following considerations: 1. If the numerator of a fraction be increased any.number of times, the fraction itself will be increased as many times; and if the denominator be diminished any number of times, the fraction must still be increased as many times. 2. If the denominator of a fraction be increased any number of times, or the numerator diminished the same number of times, the fraction itself will in either case be diminished the same number of times. 3. If the numerator of a fraction be increased any number of times, the fraction is increased the same number of times; and if the denominator be increased as many times, the fraction is again diminished the same number of times, and must therefore have its original value. The greatest common measure of the two terms of this fraction was found in page 114 to be a -y; therefore, dividing both numerator and denominator by this quantity, we obtain as our result the fraction in its lowest terms; or, 6a2+2y2 In like manner, taking the fraction 74 3 4a4-4a2b2 + 4 ab3 6a4a3b-9a2b2-3 a b 3 + 2 b 1 the greatest common measure of the two terms is found to be 2 a 2 + 2 ab —b2; and dividing both numerator and denominator by this quantity, the reduced fraction is, 2 a 2 2ab+b 2 (5.) Reduce lowest terms. Ans. 7x2 (b+c) 4a3cx-4a3dx + 24a2bcx-24a3bdx+36ab'cx-36ab2dx Tabcx3-7abdx3+ 7acsx3—7acdx3- 216* dx3 +216*cx+21bc*x+21bcdas to its 4a (a + 3b) 2 38. It frequently happens, however, that when the polynomials which form the numerator and denominator of a fraction which can be decomposed are not very complicated, we are enabled by a little practice to detect the factor and effect the reduction, without performing the operation of finding the greatest common measure, which is generally a tedious process. The results to which we called the attention of the reader, at the end of algebraic division (see page 107), will be found particularly useful in simplifications of this nature. Thus for example: (8.) 5a3+10a3b+5ab2 _ 5a(a2+2ab+b2 )_ 5a(a+b)2 = 8a2 (a+b) 8a (9) a2-2ax+x2 = (a2+ax+x2) (a—x) _ a2+ax+x2 AX (10.) = aƒ+2bx+2ax+bƒ ̄ ̄(a+b)f+2x(a+b) f+2x = (11.) = 6ac+10bc+9ad+15bd 3a(2c+3d)+5b(2c+3d) 3c(2c+3d)-(2c+3d) II. To reduce a mixed quantity to an improper fraction. 39. RULE.-Multiply the integral part by the denominator of the fraction, and to the product add the numerator with its proper sign; then the result placed over the denominator will give the improper fraction required. Thus, + 40. It is to be remarked that when a fraction has the sign whole fraction is to be subtracted, and consequently the negative sign applies to the numerator alone; and when the numerator is a polynomial, the negative sign extends to every term of the polynomial; thus, III. To reduce fractions to others equivalent, and having a common denomi nator. 41. RULE.-Multiply each of the numerators, separately, into all the denominators, except its own, for the new numerators, and all the denominators together for a common denominator. 42. RULE.-Reduce the fractions to a common denominator, add the numerators together, and subscribe the common denominator. Thus : |