the same factor can occur but once in the least common multiple, we cancel m in each of the quantities in which it is found, which is done by dividing by it. We next observe that n is a factor common to two of the remaining quantities; we therefore place it on the left, as another factor of the least common multiple, and cancel it in each of the terms in which it is found. By examining the remaining quantities, we find that x is a factor common to two of them. We then place it on the left, as another factor of the least common multiple, and cancel it in each of the terms in which it is found. We thus find that the least common multiple must contain the factors m, n, and x; it must also contain the factor mz, otherwise it would not contain all the prime factors found in one of the quantities. Hence the products, m*n\x\mz=m2nxz, contains all the prime factors of the quantities once, and does not contain any other factor; it is, therefore, the required least common multiple. Hence we have the following RULE FOR FINDING THE LEAST COMMON MULTIPLE OF TWO OR MORE QUANTITIES.-1. Arrange the quantities in a horizontal line, and divide them by any prime factor that will divide two or more of them without a remainder, and set the quotients and the undivided quantities in a line beneath. 2. Continue dividing as before, until no prime factor, except unity, will divide two or more of the quantities without a remainder. 3. Multiply the divisors and the quantities in the last line together, and the product will be the least common multiple required. OR, Separate the given quantities into their prime factors, and then multiply together such of these factors as are necessary to form a product that will contain all the prime factors in each quantity: this product will be the least common multiple required. ART. 113. Since the greatest common divisor of two quantities contains all the factors common to both, it follows, that if we divide the product of two quantities by their greatest common divisor, the quotient will be their least common multiple. Find the least common multiple of the quantities in each of the following EXAMPLES. 1. 6a2, 9ax3, and 24x3. 2. 32x2y2, 40ax3y, and 5a2x(x-y). Ans. 72a2x5. Ans. 160a2xy2(x—y). 8. x2-1, x2+1, (x−1)2, (x+1)2, x3—1, and x3+1. Ans. x10-x-xa+1. 9. 4(1-x)2, 8(1-x), 8(1+x), and 4(1+x2). Ans. 8(1-x)(1-x). 10. 3x2-11x+6, 2x2-7x+3, and 6x2-7x+2. (See Art. 113.) Ans. 6x3-25x2+23x-6. CHAPTER III. ALGEBRAIC FRACTIONS. DEFINITIONS. ART. 114. Algebraic fractions are represented in the same manner as common fractions in Arithmetic. The quantity below the line is called the denominator, because it denominates, or shows the number of parts into which the unit is divided; and the quantity above the line is called the numerator, because it numbers, or shows how many parts are taken. Thus, in the fraction, a-b b=3, c=2, and d=1, the denominator c+d shows that a unit is divided into 3 equal parts, and a-b shows that 2 of those parts are taken. ART. 115. The terms proper, improper, simple, compound, and complex, have the same meaning when applied to algebraic fractions, as to common numerical fractions. ART. 116. Every quantity not expressed under the form of a fraction, is called an entire algebraic quantity. Thus, cx-d is an entire quantity. ART. 117. Every quantity composed partly of an entire quantity and partly of a fraction, is called a mixed quantity. Thus, a+, is a mixed quantity. NOTE.-The same principles and rules are applicable to algebraic and to common numerical fractions. However, as a good knowledge of fractions is of great importance to the student, we shall present a concise demonstration of the fundamental principles and rules of operation. In these demonstrations the pupil is supposed to be acquainted with this self-evident principle: If we perform the same operations on two equal · quantities the results will be equal. ART. 118. PROPOSITION. The value of a fraction is not altered, if we multiply or divide both terms by the same quantity. Then B Q; but, from the nature of fractions, A represents a dividend, B the divisor, and Q the quotient; and by the nature of division, A=BQ. If m represents any number, then mA=mBQ; dividing these equals by mB, we have · mB Q; which proves the 1st part of the proposition. Again, take the equals A=BQ, and divide each by m, we have (Art. 73), m A m m m B Q; divide each of these equals by then m Q; which proves the 2d part of the proposition. CASE I. TO REDUCE A FRACTION TO ITS LOWEST TERMS. ART. 119. Since the value of a fraction is not changed by dividing both terms by the same quantity (Art. 118), we have the following RULE. ·Divide both terms by their greatest common divisor. OR, Resolve both terms into their prime factors, and then cancel those factors which are common. REMARK.-Instead of finding the greatest common divisor by the rule, Art. 119, it is often preferable to separate the quantities into factors by the rules for factoring (Arts. 87 to 95), and then cancel those factors common to both terms. The following examples should be solved in this manner. 17. Reduce to its lowest terms. x2+(a+c)x+ac =x(x+a)+c(x+a)=(x+c)(x+a). Also, x2+(b+c)x+bc=(x+c)(x+b); ... the fraction becomes (x+c)(x+a)_x+a 18. _ac+by+ay+bc af+2bx+2ax+bƒ' 19. 6ac+10bc+9ax+15bx 6c29cx-2c-3x = Ans. (x+c)(x+b)x+b⋅ ART. 120. Exercises in Division (see Art. 72), in which the quotient is a fraction, and capable of being reduced to lower 7. Divide 3x3-3x2-63x+135 by 3x2-2x-21. CASE II.-TO REDUCE A FRACTION TO AN ENTIRE OR MIXED QUANTITY. ART. 121. Since the numerator of the fraction may be regarded as a dividend, and the denominator the divisor, this is merely a case of division. Hence we have the following RULE. - Divide the numerator by the denominator, for the entire part, and if there be a remainder, place it over the denominator, for the fractional part. NOTE. The fractional part should be reduced to its lowest terms. |