Here, 5a2 × 2ax=10a3x; adding the numerator 3x2-a+7 to this, and we have 10a3x+3x2-a+7. Here, 4x2 × 2ac=8acx2, in adding the numerator with its proper sign; the sign - prefixed to the fraction 3ab+c signifies that it is to be taken nega tively, or that the whole of that fraction is to be subtracted; and consequently that the sign of each term of the numerator must be changed when it is 8acx2-3ab-c combined with 8acx2; hence, is -3ab-c the fraction required. Or, as =+ 2ac 2ac -3ab-c Zac (Art. 67); hence the reason of chang ing the signs of the numerator is evident. Ex. 6. Reduce x- to an improper frac tion. Ans. 5abx-a2-c To reduce an improper fraction to a whole or mixed quantity. RULE. 149. Observe which terms of the numerator are divisible by the denominator without a remainder, the quotient will give the integral part; and put the remaining terms of the numerator, if any, over the denominator for the fractional part; then the two joined together with the proper sign between them, will give the mixed quantity required. Here the operation is performed according to the rule (Art. 93), and the quotient x--x2y2 +y is the whole quantity required. 262 Here, a is the integral, and the frac tional part; therefore a 262 is the mixed quantity required. b to a mixed quantity. [required. x+a)x2-a2+b(x-a+ the mixed quantity.. x+a Here the remainder b is placed over the denominator x+a, and annexed to the quotient as in (Art. 89). Ex. 5. Reduce quantity. To reduce a fraction to its lowest terms, or most simple expression. RULE. 150. Observe what quantity will divide all the terms both of the numerator and denominator without a remainder: Divide them by this quantity, and the fraction is reduced to its lowest terms. Or, find their greatest common divisor, according to the method laid down in (Art. 141); by which divide both the numerator and denominator, and it will give the fraction required. |