LESSONS IN 



2*7 



22. Divide (* - v) x (3a + ) x 6 *<y (* - v ) x (3a + ). 



23. Divide 41U x (4 - a) x v _ I ) by (4 - a) x 41,1. 



24. Divide -40*y + 7ab* - Sahnui by - 40y + 7ab - Sahm. 

 15, I Mvide 20 (ofc + 1) - 60 (ab + 1) + 50 (ab + 1) by So. 



26. Divide 6a*+2v-3ab-bi/ + 3ao + cy + Ji 



27. Divide aab-3aa+2ab-6a -46 + 12 by b-3. 



28. Divide bb + 3be + 2co by I* + c. 



29. Divide 8aaab - bbbb by 2ab - bb. 



0. Divide xxx - SOJSB -t- Sao* - aua by a - a. 

 81. Divide 2yyy ~ Wyy + 26y - 1C by y - a 



32. Divide xxxxju - I by * - 1. 



33. Divide irxwe - 9* + 6* - 3 by 2xx + 3* - 1. 



Tho preceding rule may bo thus Hummed np : Divide every 

 part or term of the dividend by the whole divisor, and oolleo 

 the results as in addition ; the sum will bo the quotient. 



EXERCISE 10. 



1. Divide abc* by abc, and z 8 !/ 4 by **y. 



2. Divide a;in + n and x m-n each by * n . 



3. Divide ate* - aj3 + aV by at**. 



4. Divide 2a - 3a*y - 6aV by 3a. 



5. Divide 3ab3 - io a b* + SaV by 3a*b - 4a*b3. 



vide * - 13aV + 12aV by * + Sax - 4a a . 



7. Divide z* - 9xV + 12*y - 4y by z - 3ry + 2y. 



8. Divide ** - 6*" + 5z + 12* + 4 by ** - 3z - 2. 



9. Divide ** + a* by x a. 



10. Divide o - b* + 2bc - c' by o + b - c. 



11. Divide 81z + 2iz by 3*" + 2*. 



12. Divide * + a s by x* + a 8 .. 



13. Divide 10y - 23ay* + 4ay 4 by 5y4 - 4aij + a'y*. 



14. Divide 7x* - 2&I 3 + 50z 3 - 74z + 35 by z 3 - 3j^ + 5* - 7. 



15. Divide 2** - 3a?y + 2*V + y* by a: - 4y. 



16. Divide * by a;' + 2x + 1. 



17. Divide z by z - 2* + 1. 



18. Divide z* - 8* + 7 by z' - 3* + 2. 



19. Divide 5z< - Goz 3 - 2a s z - a* by z* cue + a . 



20. Divide x' - 3z*c' + 3z"r* - v* by a 3 - 3* a v -i- 3rt>* - 3 . 



21. Divide 3*" - 37z* + SSz 3 + 7z + 2 by *3 + 3i - 4x - 2. 



22. Divide 9ab + 9a"bc - 4ab + 4b*c - 9abc' - 9bc* by 3a - 2b + 3c. 



23. Divide o 3 + 3a?b + 3ab + 2b 3 + 3b s o + Sbc 3 + c 3 by a + 2b + c. 



24. Divide 4T 5 - z + 4x by 2z* + 3* + 2. 



25. Divide z - 9* + 8z l by 1 - 2z + z s . 



GREATEST COMMON MEASURE. 



109. A common measure of two or more quantities is a 

 quantity which will divide or measure each of them without 

 a remainder. [Art. 30.] Thus 2d is a common measure of 

 I2d, 6d, 8d, etc. 



110. TJie greatest common measure of two or more quantities 

 is the greatest quantity which will divide these quantities with- 

 out a remainder. Thus Qd is the greatest common measure of 

 12d and 18ci ; and 8 is the greatest common measure of 16, 24, 

 and 32. 



111. To find the greatest common measure of two given 

 quantities. 



Rule. Divide the greater of the given quantities by the less, 

 the divisor by thz remainder, and every successive divisor by its 

 own remainder, until nothing remains ; the last divisor will be the 

 greatest common measure. 



112. To find the greatest common measure of three or more 

 quantities. 



Rule. Find the greatest common measure of any two of them ; 

 then the greatest common, measure of that one and another of the 

 quantities, and so on, till all the quantities have been employed in 

 the operation; the last divisor is the greatest common measure. 



The greatest common measure of two quantities is not altered 

 by multiplying or dividing either of them by any quantity which 

 is not a divisor of the other, and which contains no factor which 

 is a divisor of the other. 



The common measure of ab and ac is o. If either be multi- 

 plied by d, the common measure of abd and ac, or ab and acd, 

 is still a. On the other hand, if ab and acd are the given 

 quantities, the common measure is a ; and if acd be divided by 

 d, the common measure of ab and ac is a. 



113. Hence, in finding the common measure by division, the 

 divisor may often be rendered more simple by dividing it by 

 aomo quantity which does not contain a divisor of the dividend. 

 Or tho dividend may be multiplied by a factor, which does not 

 contain a measure of the divisor. 



EXAMPLE. Find the greatest common measure of 6a* + 11 ax 

 + So: 1 and Go* + Vox 3J. 



Here, 6o + lax to* ) 6o + llo* + 8> ( 1 

 6o+ 7o 8* 



4<IX -f 6z 



Now dividing this remainder by 2x, wo bare 2a -f 8e for the 

 next diviBor. 



Divitor. Dividend. Quotient. 



2a + 3* ) 6a+ 7a 8a* ( 3a 

 6a* + 0a* 



2a* 3* 



2ax 3sc* 



The first remainder was divided by 20 because it is a common 

 factor of both terms of that remainder, and it cannot form a 

 factor of the common measure, not being a factor of every term 

 in the proposed quantities. As tho division of tho preceding 

 divisor by this simplified remainder leaves no remainder, there- 

 fore 2a + 3x is the common measure required. 



EXERCISE 11. 



1. Find the greatest common measure of * - far and** + 2bz + b. 



2. Find the greatest common measure of car + ** and a*e + a'x. 



3. Find the greatest common measure of 3** 24* 9 and 2if 

 - I6x - 6. 



4. Find the greatest con men measure of a* b* and a* b*a*. 



5. Find the greatest common measure of <r* 1 and <ry + y. 



6. Find the greatest common measure of ' a* and ** a*. 



7. Find the greatest common measure of a* ab - 2b* and a 1 - 3afc 



8. Find the greatest common measure of a* - ** and a* - a*z a& 

 + z. 



9. Find the greatest common measure of a 3 - ab* and a* + 2ai + b*. 



LEAST COMMON MULTIPLE. 



114. A common multiple of two or more quantities is 

 quantity which can be divided by each of them without a 

 remainder. Thus 12aZ> is a common multiple of 4a and 66 ; or 

 of 3a and 2b, etc. 



115. The least common multiple of two or more quantities its 

 the least quantity which can be divided by each of them without 

 a remainder. Thus 12a&c is the least common multiple of 4a, 

 3b, and 6c. 



116. To find the least common multiple of two or more given 

 quantities. 



Rule. Reduce the given quantities to their prime factort ; find 

 the product of tlie greatest powers of these factors, and it will fa 

 the least common multiple required. 



EXAMPLE. Find the least common multiple of (a + a:)*, 

 a 2 a*, and (a z) 2 . 



Here, tho prime factors of the quantities are (a + x)*, (a + 1), 

 (a x), and (a x) 2 ; now of these factors, which are different 

 powers of a + x and a x, the first and last contain their 

 highest powers; therefore, according to rule, (o + sr) 1 (a *)' 

 = (a* a?) 1 is the least common multiple of the quantities 

 required. 



EXERCISE 12. 



1. Find tho least common multiple of be, c, and bg. 



2. Find the least common multiple of o*b* and a*b*. 



3. Find the least common multiple of 2ob. 3bc, 4cd, 5d>, and fttf. 



4. Find the least common multiple of (a + b) J , (a* - 6*;, (a - 6) 1 . 

 and (o - b) s . 



5. Find the least common multiple of 6a, 9a*, end 4a*. 



6. Find the least common multiple of o 3 - r 1 and a 1 - *. 



7. Find the least common multiple of (* -a), (x + a), (* - ^ and 



KEY TO EXERCISES IN LESSONS IN ALGEBRA. 



EXERCISE 7. 



fcr 3-i 



7. - m - -r. 



9. L 



10. 1. 



:i. i. 



12. a + 1. 



13. b-1. 



14. ry - 1 + 2i 



15. ab + !->, 



16. 5. 



17. 1. 



6. 27 + d + -. 



