44. We shall now investigate a method of extracting the cube root of a given quantity. By actual multiplication it is seen that (a+b).(a+b).(a+b)=a3+3a2b+3ab2+b3; that is, the cube root of a3+3a2b+3ab2+b3=a+b, and we must now consider how (a+b) may be found from a3+3a2b+3ab2+b3. Now a, the first term of the root, is the cube root of the first term a3. Let a be written in the quotient; then, if a be cubed and subtracted from the original quantity, there remains 3a2b+3ab2+b3, and b may be obtained from the first term of this by dividing it by 3a2. Let 3a2 be written as the beginning of a new divisor, and b be written in the quotient. Then, if the two terms 3ab+b2 be annexed to 3a2 in the divisor, and these three terms be multiplied by b, the result will be 3a2b+3ab2+b3, which, being subtracted, leaves no remainder. Hence, the Rule for extracting the cube root of a compound quantity may be stated as follows. (a.) Arrange the terms according to the descending powers of the same letter (a). (b.) Extract the cube root of the first term; write this root (a) in the quotient, cube it and subtract. Consider the remainder as a new dividend. (3a2b+3ab2+b3). (c.) Begin to form a new divisor by squaring the quotient (a) and multiplying it by 3 (3a2). (d.) Divide the first term of the dividend by this trial divisor, and write the result (b) in the quotient. (e.) Complete the divisor by annexing to it three times the product of the two terms of the quotient (3ab), and the square of its last term (b). (f.) Multiply the divisor (3a2+3ab+b) by the last term (b) of the quotient and subtract. (g.) If there be no remainder, the root is completely extracted, but if more terms remain, proceed again according to (c), (d), (e), (f), remembering that both terms of the quotient are now to be dealt with. This process, beginning with rule (c), must be continued till the entire root is found. (2.) Find the cube root of x-6x3+21x1−44x3+63x2−54x+27 26-6x+21x4-44x3+63x2-54x+27(x2-2x+3 Working. 3(x2-2x)2 = 3x2-12x3+12x2|9x1 – 36x3+63x2 - 54x+27 +3x-2x).3+32=3x4 -12x3+21x2-18x+9 19x4-36x3+63x2−54x+27 The learner should notice carefully the way in which the second divisor is formed: the quantity (x2-2x) is regarded as a single quantity at first, and is placed within brackets; the divisor is then formed by rules (c), (d), (e), and the whole quantity is then taken out of brackets and multiplied. (3.) Find the cube roots of the following quantities : Of 26+9x+6x1 – 99x3- 42x2+441x−343; Of x+9x3y+36x7y2+84x®y3+126x3y*+126x1y3+ 84x3y®+36x2s1 Of 27a3-54a2b - 27a2c + 36ab2 + 36abc + 9ac2 8b3-12b2c -6bc2-c3; Of a9-3a8+6a7-10a6+12a3 - 12a1+10a3 - 6a2+3a-1; Of a9+9a8b2 + 27a7b1 + 3a7b + 27ab® + 27a6b3+81a5b5+3a5b2 +81a4b7+27aaba + 81a3b®+a3b3+ 81a2b3 + 9a2b5+27ab7 Of an-3a5+6a1n—7a3n+6a2n-3a2+1 Of +3x2-6x z+3xy2-12xyz+12xz2+y3-6y2z+12yz2 — 8z3. CHAPTER VII. THE GREATEST COMMON MEASURE AND THE LEAST COMMON MULTIPLE. 45. Def. ANY algebraical quantity that divides another without remainder is called a measure of it. Any algebraical quantity that divides two others without remainder is called a common measure of these two. Thus ab is a common measure of 3a2b and 6ab2. The greatest algebraical quantity that divides two others without remainder is called the greatest common measure of these two. Thus 3ab is the greatest common measure of 3a2b and 6ab2. Where the quantity consists of a single term, all its measures may be found by simple inspection. Thus the measures of 12a3b2c are 2, 3, 4, 6, 12; a, 2a, 3a, 4a, 6a, 12a; ab, 2ab, 3ab, 4ab, 6ab, 12ab; If the learner writes down all the measures, he will find that the whole number of them is 143. Write down all the measures of the following quantities : 8a2, 69ab, 15a2x2y2; ab2c+, 30a2, p2q2r2. Find the G. c. M. of 4a2x2 and 6abx. - It is evident from inspection that 2ax is the greatest quantity Examples XIV. (1.) Find the G. C. M. of the following quantities 7ab2x2 and 14a2x; a2bc2 and ab2c; -4a3 3 and 1063y3; 12xyz and -12x2y2z2 ; -48a3b2c and -36ab2c3; 10ab5c4x3y2z and 80ab2c3x1y5z6 ̧ (2.) Find the G. C. M. of the three quantities 4a2bc, 6ab2c, 5abc2 The G. C. M. of 4a2bc and 6ab2c=2abc. And G. C. M. of 4a2bc, 6ab2c and 5abc2abc. Hence, to find the G. C. M. of three algebraical expressions, find the G. C. M. of the first two, which call M, then find the G. C. M. of M and the third quantity; this will be the G. C. M. of the three expressions. This rule may be extended to find the G. C. M. of four or any number of quantities. (A proof of the rule will be given hereafter.) Find the G. C. M. of the following quantities: 3x2, 6x3, 2x; 4x3y, 6xyz, 12xyz ; 7a2b2c2, 14a2b2d2, 21a2o2d2, 21b2c2d2 15p3q2r2, 35p2q3r2, 20p2q2r3; -9abcde, - 18def, 12adf; -x2y2, x2z2, — y2x2; 4xm+2.ym+1, 12xm+1.ym+2; 8xmy2mz3m, 12x2mymz3m, 4x3my3mz3m ̧ (3.) Find the G. C. M. of a2x2 and a2x+ax2. Now a2x+ax2=ax(a+x) And the G. C. M. of a2x2 and ax(a+x)=‹ =ax. Similarly may be found the G. C. M. of the following: 3ax and (6a2 – 6ax); (4a2b−4ab2) and (2a3 — 2a2b); ax (-x) and a2x2 (a2+x2); 6a2b2, 12ab (a−b), 18ab (a+b); 46. When the quantities both contain more than one term, it will be necessary to proceed differently. The Rule in this case may be thus stated : Arrange both the quantities according to the descending powers of some letter which is common to both; make the quantity of highest dimensions the dividend, and the other the divisor; work out the division as far as possible: then make the remainder a new divisor, and the former divisor a new dividend; work out the division again as far as possible, continue the former process with the remainder and divisor until there is no remainder. The last divisor will be the greatest common measure required. If A and B represent the two quantities whose G. C. M. is required, the dimensions of A being equal to or greater than B, the process may be thus written : B) A (p C) B (q D) C (r rD Here A contains B, p times with a remainder C. It is now required to prove that D is the G. C. M. of the two quantities A and B. The proof of this rule depends upon the truth of these two propositions. 1o. If one quantity measure another, it will also measure any multiple of that other. Also, 5x measures 20x2, 30x2, 4023, which are multiples of 10x2. 2o. If one quantity measure two others, it will also measure the sum or the difference of these two. Let X measure Y, a times and Z, b times. So that Then or, and or, Example. or, Also, or, 3x2 measures 15x3 and 9x2 9x2+15x3-3x2(3+5x) 3x2 measures 9x2+15x3 15x3-9x3-3x2(5x-3) Now, by referring to the division, we see that A=pB+C B=qC+D C=rD. |