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 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 de nominator, and the result will be the simplest form of the fraction. EXAMPLES. 3ab + 6ac 1. Reduce the fraction 3ad + 12a to its simplest form. We see, by inspection, that 3 and a are factors of the numerator, hence 3ab+6ac3a (b + 2c) We also see, that 3 and a are factors of the denominator, hence 71. To reduce a mixed quantity to the form of a fraction. RULE. Multiply the entire part by the denominator of the fraction: then connect this product with the terms of the numerator by the rules for addition, and under the result place the given denominator. 72. To reduce a fraction to an entire or mixed quantity. RULE. Divide the numerator by the denominator for the entire part, and place the remainder, if any, over the denominator for the fractional part. |