The sign written before the fraction is called the sign of the fraction. Thus, if the sign of both numerator and denominator are changed, the sign of the fraction is not changed; but if the sign of either one is changed, the sign before the fraction is changed. In case the numerator or denominator is a polynomial, we must be careful, in changing the signs, to change the sign of each of its terms ( 41, 3, 4, 5). Thus, the fraction. can be written, by changing the signs of both numerator and denominator, in the a b C 129. It follows from 41, 6, 8, that if the terms of a fraction are the indicated products of two or more parentheses, the sign of the fraction will remain the same, if the signs of an even number of the parentheses be changed, but the sign of the fraction will be changed if the signs of an odd number of parentheses be changed. If the integer in the numerator of a fraction is less than its denominator the fraction is said to be a proper, or pure fraction, and if greater, an improper fraction. REDUCTION OF FRACTIONS 130. The Reduction of Fractions to their Lowest Terms. Let the line AB be divided into seven equal parts, at D, E, F, G, II, I. Now let each of these parts be subdivided into 3 equal parts. Then AB contains 21 of these subdivisions and AG contains 12 That is, the value of the fraction is not altered by multiplying both its terms by 3, and the value of the fraction is not altered by dividing both of its terms by 3. 12 21 131. The result of the previous section is a particular case of the following: Theorem I. It does not alter the value of a fraction to multiply or divide both of its terms by the same quantity. On multiplying both members of the equation by e, it becomes Hence, it follows from (1) and (2), to reduce a fraction to lower terms, divide both numerator and denominator by any factor common to both. 132. A fraction is expressed in its lowest terms if its numerator and denominator have no common factor; and therefore any fraction can be reduced to its lowest terms by dividing both numerator and denominator by their G. C. D., because it contains all the factors common to both terms of the fraction. Since in example 3 no common factor can be determined by inspection, it is necessary to determine the G. C. D. of the numerator and the denominator by the method of division. Omit the factor y from the denominator and divide. Now and (6x5xy-6 y3) y (2x-3y) = y (3x+2y) 6 x3- 11 x2y + 3xy2 x (3x-y) (2x-3y) = x (3x y) y (3x+2y) 133. When the terms of the fraction can not be readily factored, then the G. C. D. must be found by division and the terms of the fraction divided by it. EXERCISE XXIX 1. 3. 3 br 3 bc + 2 an+rbm-10 4. 21 a3b2c-9 ab3c2 15 a2b2c +3 a3b1c2 — 12 ab3c 4a2b2m¬12d+2ar+1bmc+6a2-1bm-1çn 8 ar+56m+2c2-2a+3bmc+10a"b3ca n-2n+1. n2-1 ac+bd+ad+be af +2bx+2 ax + bj 6ac9be- 5 c2 12. 14. 134. To Reduce a Fraction to an Integral or Mixed Quantity. If the degree of the numerator be equal to or greater than the degree of the denominator, the fraction may be changed to the form of a mixed or integral expression by dividing the numerator by the denominator. The quotient will be the integral part, and the remainder, if any, will be the numerator, and the divisor the denominator of the fractional part of the mixed quantity. 135. To Reduce a Mixed Expression to the Form of a Fraction. We have learned in Arithmetic that In Arithmetic, the sign connecting the fraction and the integral part of a mixed number is always +, but in Algebra, it may be + or ; so that a mixed expression may have either one of the following forms: |