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3. 25 a* - 30 alb + 19 ao63 6 ab3 + 64. 4. 1 4x + 10 m2 20 23 + 25 24 24 25 + 16 20.

5. a’ + 2 abx + (2 ac + 6*) x2 + 2 (ad + bc) 200 + (2 bd + c^) 2* + 2 cdxc5 + d.

6. a*xon 6 a 22n + 17 aRx2n 24 axen + 16 x2n-4. 7. 2 + 2 + 2C aRx - 2 2 + a-?x?. 8. 9x2m 3 aʻxcm + 25 a*

30 ax" +

+ 5 a?.

4 Find the square roots of

9. 1296, 6241, 42849, 83521. 10. 10650-24, *000576, 1, 4.

2

13 + 1 11. 117, 11:5,

75 + 1' 13 - 1 Give the values correct to four places of decimals of of :31416 15 + 12

J10 - 2 12,

3.1416 of 193

√5
12

10 + 2

48 943•

+

+

0

32.16

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Find the cube roots of 13. 8 a®boy?, 125 212/3, as + 6 a b + 12 ab2 + 863. 14. 212 + 9 x10 + 6208 99 26 42 x + 441 cm 343. 15. 20+ 3x+y + 3 xy + y 6 cxy - 3 ca? - 3 cy.

3 + 3 c^2c + 3 cʻy c.

16. + 2–3 + 3 (2c + 2–?), xy-+ 3 xʻy-' + 3 xy-'+ 1. Find the cube roots of 17. 5849513501832, 1371.330631. 18. 20-346417; -037, 4

Give the value of the following correct to four places of decimals :

35.12 + 3-03375 1 19. 380 - 7.01 9:01 94 + 32 + 1.

5 + 2

of 1:05 + 1.04

7 21. (17 + 2) ( 17 - 1), (5 + 73) (4 + 12).

108

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20. 9625 + 7:04

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22. Vil + 6 V2, V6 + 15 73.

13 23.

6

6 + 2 75 15 + 1
4
16

4 24. a: (6 c) 63 (a c) + c3 (a - b), where a = W1-2, b 1:3, and c = -0.027.

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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 22 27 and 2x^ - 11X – 63. According to the above rule, the operation is as follows: ac" - 6x

27)2 x

11 x

63(2
2 za
12 x 54

27(x + 3 aca

27

27 :: The G.C.M. is » – 9,

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Ex, 2. Find the G.C.M. of 10 203 + 31 22 63 x and 14 26 + 5122

We may tell by inspection that w is a common factor, which we therefore strike out of both, only taking care to reserve it. The quantities then become

10 x2 + 31 x 63, and 14 x2 + 51 x – 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 x 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

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 x + 51 aC 54

5 10 2 + 31 - 63) 70.00* + 2553 - 270(7 2

70 x + 217 aC 441

38 x + 171

one of

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Rejecting the factor 19 of this remainder, we have

2 x + 9)10 x + 31 x – 63(5 x - 7
x


10 x2 + 45

14 x 63
14 63

Hence, 2 x + 9 is the last divisor, and multiplying this by 2, the common measure struck out at the commencement, we find the G.C, M to be x (2 2c + 9) or 2 x2 + 9 %.

212
132

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784 20

63 3 847 aC

336

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Ex. 3. Find the G.C.M. of 3 - 7 - 33-52 +42x - 34x-21, and 2-112* + 25x + 19.0c - 49.30-21. 2:5 11 ** + 25 2,3 + 19 22 – 49 x – 21) 2° 7 2015 3 2* 5 X3 + 42 cm

34 x – 21 (2 + 4
206 11 205 + 25 + + 19 ? 49 22
4 25 28 x4 24 203 + 91 cca

21
4 205 44 204 + 100 202 + 76 cm 196 x - 84

16 * - 124 203 + 15 x + 183 x + 63
Multiplying the preceding divisor by 16, and taking the result for a dividend, we have-
16 c* – 124 23 + 15 22 + 183 x + 63)16 25 176 + + 400 203 + 304 202

336(OC
16 c5

124 204 + 15 23+ 183 m2 +

52 2* + 385 02 + 121 oca
(Multiplying this remainder by 4)

4
208 * + 1540 203 + 484

3388 x

1344( - 13
208 x4 + 1612 23 195 x2

819
72 23 + 679 x

1009 aC

525
Multiplying the preceding divisor by 9, and taking the result for a dividend, we have
72 c3 + 679 2cm

1009 oC
525)144 ac*

1116 x3 + 135 xo + 1647 x + 567( – 2 x 144 24 1358 23 + 2018 22 + 1050 x

242 23 1883 + + 597 x + 567
(Multiplying this remainder by 36)

36
8712 23 67788 cm 21492 x + 20412(121
8712 23 8.2159 22 + 122089 x + 63525

14371 22 100597 oC 43113

2379 30

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Dividing this remainder by 14371, and taking the quotient for a new divisor, we haveac – 7x - 3) - 72 dc3 +679 xc - 1009 x – 525- 72 + 175

x
72 3 + 504 + 216 2

175 2 - 1225 x 525
175 ** - 1225 x

525 :. 22 - 7 2 - 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.

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2

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. 45) be performed. Thus, let A be divided by B, with B)A(P quotient p and remainder c. Then let B be divided by C, with quotienta,

C) Bla and remainder D. Lastly, let C be qC divided by D, with quotient r, and

DCir remainder zero.

rD Then we are required to show that

0 D is the G.C.M. of A and B.

(1.) D is a common measure of A and B. Now, we have C

B qC + D, A pB + C. Hence, D is a measure of C, and therefore of qC. It is therefore a measure of qC + 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.

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