97. If only a part of the product which forms the divisor be contained in the dividend, the division must be represented by a fraction according to the direction in Art. 64, and the factors which are common to the divisor and dividend expunged. First, divide by - 3a2b, and the quotient is -5abc; this quantity is still to be divided by ≈ (Art. 95), and as a is not contained in it, the division can only be represented in the usual way; 98. If the dividend consist of several terms, and the divisor be a simple quantity, every term of the dividend must be divided by it. Thus to divide a3x2 – 5 abx3 + 6ax1 by ax2, 99. a b C (a+b+c)÷abc= + + abc abc abc 1 1 1 = + + bc ac ab When the divisor consists of several terms, arrange both the divisor and dividend according to the powers of some one letter* contained in them, (that is, beginning with the highest power and going regularly down to the lowest, or vice versa,) then find how often the first term of the divisor is contained in the first term of the dividend, and write down this quantity for the first term in the quotient; multiply the whole divisor by it, subtract the product from the dividend, and bring down to the remainder as *The operation will be shortest when that letter is chosen whose highest power in the dividend comes nearest to the highest power of the same letter in the divisor. many other terms of the dividend as the case may require, and repeat the operation till all the terms are brought down. Ex. 1. If a2 - 2ab+b2 be divided by ab, the operation The reason of this and the foregoing rule is, that as the whole dividend is made up of all its parts, the divisor is contained in the whole as often as it is contained in all the parts. In the preceding operation we inquire first, how often a is contained in a2, which gives a for the first term of the quotient; then multiplying the whole divisor by it, we have a2 ab to be subtracted from the dividend, and the remainder is ab + b2, with which we are to proceed as before. The whole quantity a2 - 2ab+b2 is in reality divided into two parts by the process, each of which is divided by a-b; therefore the true quotient is obtained. Ex. 2. a+b) ac + ad + bc + bd (c + d From this example it appears that y3-1 is divisible by y-1 without remainder, the quotient being y2+y+1. It may be shewn in the same manner that x3-a3 is divisible by x- a, the quotient being x2+ax+a2; and that a3 + a3 is divisible by x+a, the quotient being x2 - ax + a2. These results are worth remembering. Ex. 5. To divide 4ab3 + 51a2b2 + 10a1 – 48a3b – 15b1 by 4ab – 5a2 + 3b2. First arrange the terms of both dividend and divisor according to the powers of a, beginning with the highest. - 5a2 + 4ab + 3b3) 10a* — 48a3b + 51a2b2 + 4ab3 — 15b1 ( − 2a2 + 8ab — 5b2. 10a-8a3b - 6a2b2 - 40a3b+57a2b2 + 4ab3 - 40a3b+32a2b2 +24ab3 25a2b2-20ab3 – 15b4 -1 m-2 -3 1 Ex. 6. x − y) x” — yTM (x3-1 + xm−2y + x2¬3y3...+ ym−1 This division will obviously terminate without remainder for any proposed integral and positive value of m, when the quotient has reached to m terms, the last term being y1. Hence we have, - (xy) ÷ (x − y) = x2 + x3y + x2y2+xy+y'; and so on. Ex. 7. x − a) x3 − p x2 + q x − r (x2 + ( a − p) x + a2 − pa + q TRANSFORMATION OF FRACTIONS TO OTHERS OF EQUAL VALUE. 100. If the signs of all the terms both in the numerator and denominator of a fraction be changed, its value will not be altered. 101. If the numerator and denominator of a fraction be both multiplied, or both divided, by the same quantity, its value is not altered. Hence a fraction is reduced to its lowest terms by dividing both the numerator and denominator by the greatest quantity that measures them both, which quantity is called the Greatest Common Measure of the numerator and denominator. 102. A fraction which has either its numerator or denominator a simple algebraical quantity is easily reduced to lowest terms; for the greatest common divisor of the numerator and denominator is at once found by inspection, that is, by observing at sight what factors are common. 3a2bc Thus to reduce to its lowest terms, we see that a2b is the greatest 5a3b2d common measure of the numerator and denominator, therefore the fraction 3с required is But in the case of fractions having for both numerator and denominator a compound algebraical quantity the following rule is often, though not always, needed. 103. The Greatest Common Measure of two compound algebraical quantities is found by arranging them according to the powers of some letter (as in division), and then dividing the greater by the less, and the preceding divisor always by the last remainder, till the remainder is nothing; the last divisor is the greatest common measure required. pb Let a and b represent the two quantities, and b) a (p let b be contained p times in a, with a remainder c; again, let c be contained q times in b, with a remainder d; and so on, till nothing remains; let d be the last divisor, and it will be the greatest common measure of a and b. 104. The truth of this rule depends upon these two principles; c) b (q qc d) c (r rd 1st. If one quantity measure another, it will also measure any multiple of that quantity. Let a measure y by the units in n, then it will measure my by the units in mn; for since y = nx, my = = mnx (Art. 81) = mn.x, that is, a is contained mn times in my, or measures my by the units in mn. 1. If a quantity measure two others, it will measure their sum or difference. Let a be contained m times in x, and n times in y; then max and na=y; therefore a±y=ma±na = (m±n) a ; that is, a is contained m±n times in xy, or it measures a ±y by the units in m±n. 105. Now it appears from the operation (Art. 103.) that a - pb = c, and b - qc = d; every quantity therefore, which measures a and b, measures pb, and a - pb, or c; hence also it measures qc, and b qc, or d; that is, every common measure of a and h measures d. pb+c, b = qc + d, and qc, and qc + d or b; hence it It also from the division that a = appears c = rd; therefore d measures c, measures pb, and pb + c, or a. Every common measure then of |