divisor would be independent of the letter a; and it is evident that the exact division could not be performed unless the co-efficient of each term of the dividend were divisible by 3b. The exponents of the leading letter in the quotient would be the same as in the dividend. 60. Although there is some analogy between arithmetical and algebraical division, with respect to the manner in which the operations are disposed and performed, yet there is this essential difference between them, that in arithmetical division the figures of the quotient are obtained by trial, while in algebraical division the quotient obtained by dividing the first term of the partial dividend by the first term of the divisor, is always one of the terms of the quotient sought. From the third remark of Art. 45, it appears that the term of the dividend affected with the highest exponent of the leading letter, and the term affected with the lowest exponent of the same letter, may each be derived without reduction, from the multiplication of a term of the divisor by a term of the quotient. Therefore, nothing prevents our commencing the operation at the right instead of the left, since it might be performed upon the terms affected with the lowest exponent of the letter, with reference to which the arrangement has been made. Lastly, so independent are the partial operations required by the process, that after having subtracted the product of the divisor by the first term found in the quotient, we could obtain another term of the quotient by dividing by each other the two terms of the new dividend and divisor, affected with the highest exponent of a different letter from the one first selected. If the same letter is preserved, it is only because there is no reason for changing it, and because the two polynomials are already arranged with reference to it; the first terms on the left of the dividend and divisor being sufficient to obtain a term of the quotient; whereas, if the letter is changed, it would be necessary to seek again for the highest exponent of this letter 61. Among the different examples of algebraic division, there is one remarkable for its applications. It is expressed thus: The difference between the same powers of any two quantities is always divisible by the difference between the quantities. Let the quantities be represented by a and b; and let m denote any positive whole number. Then, am- bm will express the difference between the same powers of a and b, and it is to be proved that am bm is exactly divisible by a - b. If we begin the division of am bm by a - b, we have Dividing am by a the quotient is am-1, by the rule for the exponents. The product of a - b by am-1 being subtracted from the dividend, the first remainder is am-1b-bm, which can be put under the form b(am-1 ·bm-1). Now, if the factor (am-1-bm-1) of the remainder, be divisi If the difference of the same powers of two quantities be divisible by the difference of the quantities, then, the difference of the powers of a degree greater by unity is also divisible by it. But by the rules for division, we have Hence, we know, from what has just been proved, that a3- 63 is divisible by ab, and from that result we conclude that at b4 is divisible by ab, and so on, until we reach any exponent at m. CHAPTER III. OF ALGEBRAIC FRACTIONS. 62. ALGEBRAIC fractions are to be considered in the same point of view as arithmetical fractions; that is, a unit is supposed to be divided into as many equal parts as there are units in the denominator, and one of these parts is supposed to be taken as many times as there are units in the numerator. Thus, in the fractional expression a+b c + ď a given unit is supposed to be divided into as many equal parts as there are units in c+d, and as many of these parts are taken, as there are units in a + b. The rules for performing Addition, Subtraction, Multiplication, and Division, are the same as in arithmetical fractions. Hence, it will not be necessary to demonstrate these rules, and in their application we must follow the methods already indicated in similar operations on entire algebraic quantities. 63. Every quantity which is not expressed under a fractional form, is called an entire algebraic quantity. 64. An algebraic expression, composed partly of an entire quantity and partly of a fraction, is called a mixed quantity. 65. When the division of two monomial quantities cannot be performed exactly, it is indicated by means of the known sign, and in this case, the quotient is presented under the form of a fraction, which we have already learned how to simplify (Art. 51). With respect to polynomial fractions, the following are cases which are easily reduced. which can be put under the form (Art. 48): and by suppressing the common factors, a (a - b), the result is In the particular cases examined above, the two terms of the fraction are decomposed into factors, and then the factors common to the numerator and denominator are cancelled. Practice teaches the manner of performing these decompositions, when they are possible. But the two terms of the fraction may be complicated polynomials, and then, their decomposition into factors not being so easy, we have recourse to the process for finding the greatest common divisor, which is explained at page 300. CASE I. 70. To reduce a fraction to its simplest form. RULE. I. Decompose the numerator and denominator into factors, as in Art. 48. II. Then cancel the factors common to the numerator and denominator, and the result will be the simplest form of the fraction. |