153. If either numerator or denominator is a polynomial, care must be taken, on changing its sign, to change the sign of each of its terms. Thus, the fraction a-b c-d by changing the signs of both numerator and denominator, can be written in the form 154. It follows from Art. 151 that the fraction written in any one of the forms (-a)b (-a)(-b) (-a)(-b) etc.; etc. From which it appears that If the terms of a fraction are composed of factors, the signs of any even number of factors may be changed without altering the value of the fraction. But if the signs of any odd number of factors are changed, the sign before the fraction is changed. 155. Since a fraction is an expression of division, we have the following rule: Divide the numerator by the denominator. 6x2-15x-2 1. Reduce to a mixed quantity. 3 x Dividing each term of the numerator by the denominator, A remainder whose first term will not contain the first term of the divisor, may be written over the divisor in the form of a fraction, and added to the quotient. Thus, the result is Or, since the sign of each term of the numerator may be changed, if at the same time the sign before the fraction is 5. x+2y a3- a2-a-2 8. 12x2 - 8x +7. x+y 4x-1 156. The operation being the converse of that of Art. 155, we have the following rule: Multiply the integral part by the denominator; add the numerator to the product when the sign before the fraction is +, and subtract it when the sign is ; and write the result over the denominator. a+b_a2-b2 -5 _ (a + b) (a – b) – (a2 – b2 – 5) = - - a-b a-b Note. If the numerator is a polynomial, it will be found convenient to enclose it in a parenthesis, when the sign before the fraction is -. 157. 1. Reduce DENOMINATOR. 5cd 3 mx 3a2b' 2 ab2' and 3ny to equivalent frac 4 ab tions having the lowest common denominator. The lowest common denominator is the lowest common multiple of 3 a2b, 2 ab2, and 4 ab, which is 12ab2. By Art. 147, both terms of a fraction may be multiplied by the same quantity without altering its value. Hence, 9 bny 12 a3b2 Multiplying both terms of 3 ny by 36, we have 4ab Therefore the required fractions are 20 abcd 18 a2mx , , It will be observed that the terms of each fraction are multiplied by a quantity which is obtained by dividing the lowest common denominator by its own denominator. Hence the following rule : Find the lowest common multiple of the given denominators. Divide this by each denominator separately, multiply the corresponding numerators by the quotients, and write the results over the common denominator. Note. Before applying the rule, each fraction should be in its lowest terms. EXAMPLES. Reduce the following to equivalent fractions having the lowest common denominator: |