3. 25 a* - 30 ab + 19 a?b% - 6 ab3 + 64. 4. 1 - 4x + 102 – 2023 + 25 ** - 24.200 + 16 2. 5. a” + 2 abx + (2 ac + 62) 202 + 2 (ad + bc) 208 + (2 bd + c) ** + 2 cd.2c5 + ddce. 6. a*22n – 6 ao.c2n –1 + 17 a 221 – ? - 24 axen – 3 + 16 221–4. 7. 2? + 2 + 2C?, a-x^2 - 2 + a-?co. 8. 9 cm - 3 aşam + 25 a- 30 axm + 9 + 5 ad. Find the square roots of — 9. 1296, 6241, 42849, 83521. 10. 10650-24, .000576, 1, 45. 2 13 + 1 11. 117, 11.5, Jo ft Nai Give the values correct to four places of decimals of— f of -31416 15 + 12 + V10 — 2, 3.1416 of 193 15 - 12 * 110 + 20 Find the cube roots of13. 8 a?boyl?, 125 x??y?, ax + 6 aʼ6 + 12 ab? + 863. 14. 2012 + 9 2010 + 6 28 – 99 228 – 42 2* + 441 22 – 343. 15. 2018 + 3x*y + 3 xy + y3 - 6 czy - 3 cæ2 - 3 cy? + 3 cc + 3 c°g - c. . 16. 2x3 + 3 + 3 (3C + 2–?), x*y–3 + 3 x*y–% + 3 xy-1 + 1. Find the cube roots of 17. 5849513501832, 1371.330631. 18. 20-346417; •037, 130485. Give the value of the following correct to four places of decimals :10 V5.12 + 3:03375 1 3/80 - 501 94 + 32 + 1. on 7.625 + 3/04 15 + 2 7:05 + 1.04 of 7 . 21. (17 + 2) ( 17 – 1), (5 + 13) (4 + 712). 22. Vil + 6 12, 7 6 + 15 13. 23. 3 6 + 2 15 15 + 1 7164 24. a? (6 – c) – 63 (a – c) + c (a - b), where a = 11-2, b = - V.3, and c = - %027. CHAPTER IV. GREATEST COMMON MEASURE AND LEAST COMMON MULTIPLE.. Greatest Common Measure. 44. In Arithmetic (page 24) we defined the G.C.M. of two or more numbers as their highest common factor. In Algebra the same definition will suffice, provided we understand by the term highest com:non factor, the factor of highest dimensions (Art. 18). This, it need hardly be remarked, does not necessarily correspond to the factor of highest numerical value. 45. To find the G.C.M. of two quantities. · RULE.—Let A and B be the quantities, of which A is not of lower dimensions than B. Divide A by B, until a remainder is obtained of lower dimensions than B. Take this remainder as a new divisor, and the preceding divisor A as a new dividend, and divide till a remainder is again obtained of lower dimensions than the divisor; and so on. The last divisor is the G.C.M. Before giving the general theory of the G.C.M. we shall work out a few examples. Ex. 1. Find the G.C.M. of x2 - 6x – 27 and 2x – 11x – 63, According to the above rule, the operation is as follows:co – 6 x - 27)2 c° - 11 = - 6302 '22* – 12 x – 54 2 - 9) 22 - 6 x – 27(30 + 3 aca – 9 x 3x – 27 3x – 27 ;: The G.C.M. is a – 9, Ex. 2. Find the G.C.M. of 10 203 + 31 32 – 63 x and 14.28 + 51 202 - 54 x. We may tell by inspection that x is a common factor, which we therefore strike out of both, only taking care to reserve it. The quantities then become 102c2 + 31 x – 63, and 14 m2 + 51 2 – 54. We may now proceed according to rule, taking the former as divisor. We see, however, that the coefficient of the first term of the dividend is not exactly divisible by the coefficient of the first term of the divisor. Multiply therefore (to avoid fractions) the dividend by such a number as will make it so divisible, viz., by 5. This will not affect the G.C.M., as 5 is not a factor of the first expression, viz., 10 x2 + 31 2 – 63. It may as well be here mentioned that the G.C.M. of two quantities cannot be affected by the multiplication or division of one of the quantities by any quantity which is not a measure of the other. We shall, for a similar reason, reject certain factors or introduce them into any of the remainders or dividends during the operation. (See Art. 47). 14 x2 + 51 x – 54 102c + 31 2 – 63)70 ac + 255 x – 270(7 70 2a + 217 2 – 441 38 x + 171 Rejecting the factor 19 of this remainder, we have 2 x + 9)10 m2 + 31 x – 63(5 x - 7 10 cm + 45 x - 14 2 - 63 - 14 X - 63 Hence, 2 x + 9 is the last divisor, and multiplying this by 3, the common measure struck out at the commencement, we find the G.C.M to be » (2 x + 9) or 2 202 + 9 x. Ex. 3. Find the G.C.M. of 28 – 7000 – 3 **-529 +42x2 – 34x-21, and x3 – 1124 + 25x8 + 19x - 49x-21. 203 - 11 24 + 25 23 + 19 22 – 49 x – 21) 26 – 7 2015 - 320* 5 203 + 42 x - 34 20 – 21 (a + 4 26 – 11 200 + 25 2* + 19 x? – 49 x - 21 x 4 26 – 28 c* - 24 x3 + 91 m2 - 13x – 21 4 205 - 44 204 + 100 203 + 76 m2 – 196 x - 84 16 2* – 124 2.3 + 15 m2 + 183 XC + 63 Multiplying the preceding divisor by 16, and taking the result for a dividend, we have16 24 - 124 23 + 15 22 + 183 x + 63)16 oct - 176 c* + 400 203 + 304 22 - 784 x – 336(OC 16 205 – 124 c* + 15 23 + 183 x2 + 632 - 52 c* + 385 203 + 121 22 – 847 2 – 336 (Multiplying this remainder by 4) - 208 2* + 1540 203 + 484 34 – 3388 x – 1344( - 13 - 208 x* + 1612 203 - 195 x2 - 2379 - 819 - 72 243 + 679 c2 - 1009 x – 525 Multiplying the preceding divisor by 9, and taking the result for a dividend, we have .- 72 203 + 679 22 - 1009 x – 525)144 c4 - 1116 2c3 + 135 m2 + 1647 x + 5671 – 2 x 144 204 – 1358 x3 + 2018 c2 + 1050 a 242 2c – 1883 cm + 597 3 + 567 36 14371 ac – 100597 x – 43113 Dividing this remainder by 14371, and taking the quotient for a new divisor, we have2c2 – 7 20 – 3) – 72 23 + 679 – 1009 3 – 525( - 72 x + 175 - 72 203 + 5042c2 + 216 x 1752 - 1225 x – 525 175 x – 1225 x – 525 :: 202 – 7 x – 3 is the G.C.M. It will be seen that we have introduced and rejected factors during the operation in order to avoid fractional coefficients. This, as will be seen from the general theory, will not affect the result, provided that no factor thus introduced or rejected is a measure of the corresponding divisor or dividend, as the case may be. 90 Theory of the Greatest Common Measure. 46. Let A and B be the two algebraical quantities, and the operation as indicated by the rule (Art. 25) be performed. Thus, let A be divided by B, with BAP quotient p and remainder C. Then pВ let B be divided by C, with quotient q, C) Bla and remainder D. Lastly, let C be divided by D, with quotient r, and DC(r remainder zero. Then we are required to show that D is the G.C.M. of A and B. (1.) D is a common measure of A and B. Now, we have C = rD, B = q C + D, A = pB + C. Hence, D is a measure of C, and therefore of aC. It is therefore a measure of gC + D or B. Hence, also, D is a measure of pB, and since it is also a measure of C, it must be a measure of pB + C or A. But we have shown it to be a measure of B. Hence, D is a common measure of A and B. (2.) D is the G.C.M. of A and B. For every measure of A and B will divide A - pB or C; and hence every measure of A and B will divide B - qC or D. Now, D cannot be divided by any quantity higher than D, and, therefore, there cannot exist a measure of A and B higher than D. Hence, D is the G.C.M. of A and B. |