ALGEBRA. 



and the sum of the negative co-efficients, 

 prefix the sign of the greater sum, and 

 annex the common letters. 



Ex. 7. 3 a + 4 b c r* + 10 x 25 



5a'-f6Ac+2e' 15x+ 44 



4 a' 9 b c 10<-> +21 .r 90 



Ex. 6. From 4 a 3 A-f 6 c 11 

 take 10x4- a 15 2y 

 Ans. 3 a 3 A-j-6 c 10x 



Ex. 7. From a x~s b x*-\-x 

 take p x* q x*+r x 



Ans. 



'be 9 e> 4- 16 x - 71 An3 - /.x3 A g ..rH-l r-J 



E\.S. 4ac 15 b d 4- x ax 



11 ac+ 7 A' 19<x4-4a 



41 a 1 4- 6bd 7de2a 



A.] ~ 



^la 1 9 1 </-t-7A l 18fx 



Ex. 9. / a-? 9 x* r x 

 ax? Ax* x 



Ans. /Ht-a 



r-f-l.x 



In this example, the co-efficients of x 

 and its powers are united ; p-\- a . xi=p 

 r?-|-ax' ; also ?+A x l = q x* b 

 x 1 , because the negative sign affects the 

 whole quantity under the vmculum ; and 



i -- 1 . x= rx x 



SUBTRACTION-. 



fiul'trartifin, or the taking away of one 

 quantity fromunot/ier, it performed by chang- 

 ing tin' sign oftlte quantity to be subtracted, 

 and then adding it to the other, by the rules 

 laid {/overt ';i the last article. 



Ex. 1. From 2 A x take c y, and the dif- 

 tVrence is properly represented by 2 b x 



r y ; because the prefixed to c y 

 shews that it is to be subtracted from the 

 other ; and 2 l> -r c y is the sum of 2 b x 

 uiul c y. 



Ex. 2. Again, from 2 b x take cy, and 

 the difference is 2 A x-\-c y ; because 2 b x 

 =2 A x -|-c y cy, take ;iu :iy c y from 

 these equal quantities, and the differences 

 \\ill beeqmtl) /. e. the difference between 

 2 A x and r y is 2 b r-\-c y, the quantity 

 which arises from adding HJ-cy to 2 A x. 



Ex. 3. From a -f b 



take a A 



Ans.* + 2 A 



Ex. 4. From 6 a 12 A 

 take 5 H 10 A 

 Ans. ll a 2 A 



Ex. 5. From 5 >-f-4 a I, 6 x y 



take ll fl '-j-6a b 4xy 



Ans. _ 6 rt 2 ab 2 x ~ 



VOL.1. -- - - 



In this example the co-efficients are 

 united; a p . ri is equal to A x 1 9 x' ; 



A ? . x* is equal to A x q r 1 ; and 

 i r . x=.v r x. 



JCt'LTiritCATlOlt. 



The multiplication of simple algebrai- 

 cal quantities must be represented ac- 

 cording to the notation already pointed 

 out. 



Thus, a X b, or a A, represents the pro- 

 duct a multiplied by A ; a b c, the pro- 

 duct of the three quantities, a, A, and c . 



It is also indifferent in what order they 

 are placed, a X b and A X a being equal. 



To determine the tign of the product, 

 observe the following rule. 



If the tmtltipb'rr and multiplicand have the 

 same sign, the product is positive; if they 

 Aave different signs, it is negative. 



1. _j_'a x 4- A = a A ; because in this 

 case a is to be taken positively A times ; 

 therefore the product a b must be posi- 

 tive. 



2. a X 4- A = a A ; because a is 

 to be taken A times ; that is, we must take 



ab. 



3. 4aX A = ab; for a quantity is 

 said to be multiplied by a negative num- 

 ber A, if it be subtracted A times ; and 

 a subtracted b ti.nes is ab. 



4. ox A=4oA. Here a is to 

 be subtracted A times; that is, a A is to 

 be subtracted ; but subtracting a b IB 

 the same as adding 4~ ^ ! therefore we 

 have to add + a ^- 



The 2" 1 and 4 th cases may be thus prov- 

 t .,l . a it=o, multiply both sides l>y A, 

 and tOjfetherwith aXAmtut be equal 

 , -, or nothing ; therefore, a mul- 



: by b mus : . 



which when added to ab makes the sum 

 notiiing. 



Again, a n=o ; multiply both side* 

 by b, tlvn a f together with oX 

 A must be =o ; therefore a X * 



. 



If the quantities to be multiplied hare 

 co-efficients, these must be multiplied to- 

 O 





