mainder. Thus 12a2 is the least common multiple of 3a2 and 4a. 102. It is obvious that the least common multiple of two or more quantities must contain all the factors of each of the quantities, and no other factors. Hence, when the given quantities can be resolved into prime factors, the least common multiple may be found by the following RULE. Resolve each of the quantities into its prime factors; take each factor the greatest number of times it enters any of the quantities; multiply together the factors thus obtained, and the product will be the least common multiple required. EXAMPLES. 1. Find the least common multiple of 9x2y and 12xy2. Resolving into factors, we have 9x2y=3x3xxy, and 12xy2=3×2×2xyy. The factor 3 enters twice in the first quantity, also the factor 2 enters twice in the second; x twice in the first, and y twice in the second. Hence the least common multiple is 2×2×3×3xxyy, or 36x2y2. 2. Find the least common multiple of 4a2b2, 6a2b, and 10a3x2. We have 4a2b2-2x2aabb, 6a2b=2×3aab, 10а3х2=2х Бааахх. Hence the least common multiple is 2×2×3×5aaabbxx, or 60a3b2x2. 3. Find the least common multiple of a2x-2abx+b2x and a2y-b3y. Here we have a2x-2abx+b2x=(a—b) (a—b)x, a2y—b2y=(a+b) (a—b) y. Hence the least common multiple is (a - b) (a —b) (a+b)xy, or a3xy—ab2xy — a2bxy+b3xy. 6a2x2. 4. Find the least common multiple of 5a2b2, 10ab3, and 2abx. Ans. 10a2b3x. 5. Find the least common multiple of 3ab2, 4ax2, 562x, and Ans. 60a2b2x2. 6. Find the least common multiple of x2-3x+2 and x2-1. Ans. (x+1)(x-1) (x−2), or x3— 2x2 −x+2. 7. Find the least common multiple of a3x+b3x and 5a2—5b2. Ans. 5x(a+b)(a—b) (a2 — ab+b2), 103. When the quantities can not be resolved into factors by any of the preceding methods, the least common multiple may be found by applying the following principles: If two polynomials have no common divisor, their product must be their least common multiple; but if they have a common divisor, their product must contain the second power of this common divisor. Their least common multiple will therefore be obtained by dividing their product by their greatest common divisor. Hence, to find the least common multiple of two quantities, we have the following RULE. Divide the product of the two polynomials by their greatest common divisor; or divide one of the polynomials by the greatest common divisor, and multiply the other by the quotient. EXAMPLES. 1. Find the least common multiple of 6x2-x-1 and 2x2+3x-2. The greatest common divisor of the given quantities is 2x1. Hence the least common multiple is (6x2-x-1) (2x2+3x-2) 2x-1 or (2x2+3x-2) (3x+1). 2. Find the least common multiple of x3-1 and x2+x−2. Ans. (x3-1)(x+2). 3. Find the least common multiple of x3-9x2+23x−15 Ans. (x3-9x2+23x-15) (x-7). and x2-8x+7. 104. When there are more than two polynomials, find the least common multiple of any two of them; then find the least common multiple of this result, and a third polynomial; and so on to the last. 4. Find the least common multiple of a2+2a−3, a2—1, and a—1. Ans. (a2-1) (a+3). 5. Find the least common multiple of 4a2+1, 4a2—1, and 2a-1. Ans. 16a+-1. 6. Find the least common multiple of a3—a, a3+1, and a3-1. Ans. a (a-1). 7. Find the least common multiple of (x+2a)3, (x—2a)3, and x2-4a2. Ans. (x2-4a2)3. CHAPTER VII. FRACTIONS. 105. A fraction is a quotient expressed as described in Art. 71, by writing the divisor under the dividend with a line between them. Thus is a fraction, and is read a divided by b. 106. Every fraction is composed of two parts: the divisor, which is called the denominator, and the dividend, which is called the numerator. 107. An entire quantity is an algebraic expression which has no fractional part, as a2-2ab. An entire quantity may be regarded as a fraction whose denominator is unity. Thus, a2= = a2 1 108. A mixed quantity is an expression which has both entire and fractional parts. Thus a2+ is a mixed quantity. b 109. General Principles of Fractions.-The following principles form the basis of most of the operations upon fractions: и 1st. In order to multiply a fraction by any number, we must multiply its numerator or divide its denominator by that number. Thus the value of the fraction abis b. If we multiply the numerator by a, we obtain a2h, or ab; and if we divide the denominator of the same fraction by a, we obtain also ab; that is, the original value of the fraction, b, has been multiplied by a. a 2d. In order to divide a fraction by any number, we must divide its numerator or multiply its denominator by that number. Thus the value of the fraction a2b is ab. If we divide the a numerator by a, we obtain ab, or b; and if we multiply the denominator of the same fraction by a, we obtain ab, or b; that a2, is, the original value of the fraction ab has been divided by a. 3d. The value of a fraction is not changed if we multiply or divide both numerator and denominator by the same number. 110. The proper Sign of a Fraction.-Each term in the numer ator and denominator of a fraction has its own particular sign, and a sign is also written before the dividing line of a fraction. The relation of these signs to each other is determined by the principles already established for division. The sign prefixed to the numerator of a fraction affects merely the dividend; the sign prefixed to the denominator affects merely the divisor; but the sign prefixed to the dividing line of a fraction affects the quotient. The latter sign may be called the apparent sign of the fraction, while the real sign of the fraction is the sign of its numerical value when reduced. The real sign of a fraction depends not merely upon its apparent sign, but also upon the signs of the numerator and denominator. From Art. 73, it follows that Also, since a minus sign before the dividing line of a fraction shows that the quotient is to be subtracted, which is done by changing its sign, it follows that Hence we see that of the three signs belonging to the numer |