simple quantity common to all the terms; and, to avoid fractions in the quotients, any dividend may be multiplied by any simple quantity. Care must be taken, however, that no factor rejected be common to both of the original quantities. If there be such a common factor, it is better to divide both quantities by it, and afterwards to introduce it as a factor into the common measure. The Greatest Common Measure of three or more quantities may be found, by first finding the Greatest Common Measure of any two of them, then of that and a third, and so on. The following example will show the method of applying the above rule. To find the Greatest Common Measure of 8x3+6x2-4x-3, and 12x3 +5x2+x+3. 8+ 6 4 3 12 +5 +1+3 8-14-15 20 + 11 -3 2 -9 Writing out the coefficients as in ordinary Division, since the quantities are of the same dimension, we may divide either by the other. Doubling the latter to make the first term divisible by 8, and di 40 + 22 24+10+2 +63 6 -8 40 -70-75 23) 92 +69 4 + 3 or 4x+3 -8+14 +15-1-5 6 20 +15 viding, we get 3 for quotient, and the remainder is-8+14 +15; then dividing the divisor by this, we first get for quotient 1, with the remainder 20+11-3, and doubling this to make the first term divisible by -8, we get for the next part of the quotient -5, with the remainder 92+69, or by division by 23, 4+3. Lastly, Dividing the last divisor by this remainder, we get -2+5 for quotient, and no remainder; hence 43 are the coefficients of the Greatest Common Measure, which is therefore 4x+3.* It is usual in operations such as the above to write the dividend always on the right-hand side of the divisor; but this serves no good purpose, as the work is just as easy when the position of each is reversed. * Since the first sign in the divisor, 81415, is all the signs might have been changed, or it might have been multiplied by-1. B EXERCISES. Find the Greatest Common Measure of the terms of the following fractions, and reduce them to their lowest terms:- simple quantity common to all the terms; and, to avoid fractions in the quotients, any dividend may be multiplied by any simple quantity. Care must be taken, however, that no factor rejected be common to both of the original quantities. If there be such a common factor, it is better to divide both quantities by it, and afterwards to introduce it as a factor into the common measure. The Greatest Common Measure of three or more quantities may be found, by first finding the Greatest Common Measure of any two of them, then of that and a third, and so on. The following example will show the method of applying the above rule. To find the Greatest Common Measure of 8x3+6x2-4x-3, and 12x3 +5x2+x+3. 8+ 6 4 3 12 +5 +1 +3| 8 -14-15 20 + 11 3 2 2 24+10+2 +63 Writing out the coefficients as in ordinary Division, since the quantities are of the same dimension, we may divide either by the other. Doubling the latter to make the first term divisible by 8, and di 40+22 6 40 -70-75 23) 92 +69 or 4 + 3 -8+14 +15 |—1. —8—6 20 +15 20+152+5 viding, we get 3 for quotient, and the remainder is-8+14 +15; then dividing the divisor by this, we first get for quotient1, with the remainder 20+11-3, and doubling this to make the first term divisible by -8, we get for the next part of the quotient -5, with the remainder 92+69, or by division by 23, 4+3. Lastly, Dividing the last divisor by this remainder, we get -2+5 for quotient, and no remainder; hence 43 are the coefficients of the Greatest Common Measure, which is therefore 4x + 3.* It is usual in operations such as the above to write the dividend always on the right-hand side of the divisor; but this serves no good purpose, as the work is just as easy when the position of each is reversed. * Since the first sign in the divisor, - 8+14 + 15, is all the signs might have been changed, or it might have been multiplied by-1. B |