44. CASE II. To reduce improper fractions to whole or mixed quantities. RULE. Divide the numerator by the denominator, as far as an integral quotient can be obtained; then write the remainder for the numerator, and the former denominator for the denominator of the fraction. Here 12a divides by 4a, and gives an integral quotient, so also does 4a divided by 4a; but 3c will not give an integral quotient when divided by 4a, therefore it is written. as a fraction at the end with its proper sign. 45. LEMMA. To find the greatest common measure of two algebraic quantities; that is, the greatest quantity that will divide them both without leaving a remainder. RULE. Arrange the quantities according to the powers of the same letter, as in division, then divide that which contains the highest power of that letter by the other, and the last divisor by the remainder continually, till there be no remainder, the last divisor used is the greatest common measure. NOTE 1. If all the terms of one of the quantities be divisible by any simple quantity which does not divide every term of the other, since it cannot form a part of the common measure, it may be struck out of all the terms by dividing by that simple quantity before the general division is performed. NOTE 2. If the coefficient of the leading term of the dividend be not divisible by the coefficient of the leading term of the divisor, every term of the dividend may be multiplied by such a number as will make its coefficient divisible by that of the divisor without remainder; by this means fractions are avoided. NOTE 3. Place the quantities in two columns, so that the one which is to be the divisor may stand in the left hand column, and the other in the right; then divide the right hand column by the left, and place the quotient after the right hand column; then divide the left hand column by the remainder, (reduced, if necessary, by Note 1), and place the quotient before the left hand column; proceed in this manner till there be no remainder. EXAMPLE. Required the greatest common measure of the quantities x2 +2ax+a2, and x3-ax. 46. The process of finding the greatest common measure can often be much more elegantly performed, by reducing the quantities into factors by the theorems, at the end of multiplication and division; then the product of all the factors that are common to both quantities is the greatest common measure sought. The above example wrought in this way would be as under, x3—a2x—x(x2—a2)=x(x+a)(x—a), Thence x+a being the only factor common to both, is the greatest common measure. 1. Find the greatest common measure of a1-x1, and a3-2a2x+ax2-2x3. Ans. a2+x2. 2. Find the greatest common measure of x+y3, and x2+x3y—xy3 —y4. Ans. x+y. 3. Find the greatest common measure of 48x2 +16x-15, and 24x3-22x2+17x-5. Ans. 12x-5. 4. Find the greatest common measure of x-ao, and x4+4x3a+6x2a2+4xa3+a+. Ans. x+α. 47. CASE III. To reduce fractions to their lowest terms. RULE. Divide both numerator and denominator by their greatest common measure, and the quotients will be the fraction in its lowest terms. EXAMPLE. Reduce x3-a2x x3-3x2a+3xa2 to its lowest terms, Here it will be found, that x-a is the greatest common measure; therefore we have the following:x3-a2x ÷(x-a) 22+ax x3-3x2a+3xa2—a3 ÷ (x—a) = Ans. x2—2ax+aa• ax+x2 a+x 1. Reduce 3bx-cx 3b x+c a+2x Ans. a2-2ax* m*—n1 4. Reduce to its lowest terms. m2+n2 m1-2m2n2+n1 Ans. 5. Reduce to its lowest terms. 8x3-27a3 48. CASE IV. To reduce fractions with different denominators, to equivalent ones having a common denomi nator. RULE 1. Multiply both numerator and denominator of each fraction by the product of the denominators of all the others; the resulting fractions will have a common denominator. RULE 2. Multiply the terms of each fraction by the quotient arising from dividing the least common multiple of all the denominators by its denominator; the resulting fractions will have the least common denominator. 49. NOTE. The least common multiple of several quantities is the least quantity that is divisible by each of them, without leaving a remainder. It may be found by resolving each of the quantities into its simplest factors, and then multiplying together all the separate factors that appear in the whole, using the highest power of each that appears in any one quantity. For example, let us find the least common multiple of (x3—a3)=(x2+xa+a2)(x—a), (x2—2ax+a2), =(x—a)3, x2—a2=(x+a)(x—a), it will be the product of (x2+xa+a3) (x-a)(x+a), because these are all the different factors which ap pear in the whole, and (x-a) is used in the second power, because it appears in that power in one of them. By RULE 2. The least common multiple of all the denominators is 12ax; therefore the fractions reduced to their least common denominator are 3x 9x2 3x b 12ax' 4a 4a3 3x 4a 12ax* Ans. 5x 3x 46 and 5bx(a2-b2) 3a2bx-3ab2x 4a2b2+4ab3 3a3x-3ab2x ab(ab) ab(a"-b2) ab(a3-b2) ab(a-b2) 7a 3x 4. Reduce to their least common denominator a+x' a -I ADDITION OF ALGEBRAIC FRACTIONS. 50. RULE. Reduce the fractions to a common denominator; then add their numerators, and under the sum write their common denominator; the fraction so formed will be the fraction required. |