ALGEBRA. 



and the sum of the negative co-efficients, Ex. 6. From 4 a 3 6-f-6 c 11 

 prefix the sign of the greater sum, and take 10 x+ a 15 2 / 



annex the common letters. 



Ans. 3 a 3 6-f6 c 10 x-f2 y-j-4 



25 



Ex. 7. From a a:3 



take /> .r? g x*--r x 



Ans. a p . x? b q . x*+I 



Ex.7. 3 



4 a* 9 b c IQe* + 21 x 90 

 Ans.^^6 a 1 -f b c 9e* + 16 x 71 



Ex. 8 4 a c - 15 bd + e ar - fl * In Aisjxample the co-efficients are 



llac-f 76* \9ex-\-4, ax united ; a p . 373 is equal to bx* gx>\ 



41 ' +- 6b d 7 de 2 ax b g . x 1 is equal to b x* q x* ; and 



A. I5a^4la^ 9*d+7t>* I8ex 7de ax i - r . 0:1=0: r a:. 



Ex. 9. px^ qx*rx 

 axi b x* x 



Ans. />-f-a . o:3 g -j-6 . x 1 r+1. x 



In this example, the co-efficients ofo: 

 and its powers are united ; p+a. xi=p 

 ,r3 -}- a x> ; also g-\-b . x* = q x 1 b 

 x 1 , because the negative sign affects the 

 whole quantity under the vinculum ; and 



r 1 . x r x x. 



SUBTRACTION. 



Subtraction, or the taking away of one 

 quantity from another, is performed by chang- 

 ing the sign of the quantity to be subtracted, 

 and then adding it to the other, by the rules 

 laid doion in the last article. 



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

 ference is properly represented by 2 b x 



c y / because the prefixed to c y 

 shews that it is to be subtracted from the 

 other ; and 2 b x c y is the sum of 2 b x 

 and c y. 



Ex. 2. Again, from 2 bx take c y, and 

 the difference is 2 b x-\-c y , because 2 b x 

 =2 b x~\-c y c y, take away c y from 

 these equal quantities, and the differences 

 will be equal; i. e. the difference between 

 2 b x and cy is 2 b x + c y, the quantity 

 ^hich arises from adding -f- c y to 2 b x. 



Ex. 



3. From a + b 



take a b 



Ans. * 4- 2 b 



Ex. 4. From 6 a 12 b 

 take 5 a 10 b 

 Ans. 11 p-ITTA 



Ex. 5. From 5 a*-f-4 a b & x y 



{take 11 aH-6 a b 4 x y 

 Ans. 6 a* 2 a b 2 x ?/ 

 VOL. I. " 



MULTIPLICATION. 



The multiplication of simple algebrai- 

 cal quantities must be represented ac- 

 cording to the notation already pointed 

 out. 



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

 duct a multiplied by b ; a be, the product 

 of the three quantities, a, b, and c. 



It is also indifferent in what order tlvey 

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



To determine the sign of the product, 

 observe the following rule. 



If the multiplier and multiplicand have the 

 same sign, the product is positive ; if they 

 have different signs, it is negative. 



1. + aX -\- b = ab ; because in this 

 case a is to be taken positively b times ; 

 therefore the product a b must be posi- 

 tive. 



2. ax-M= ob; because a 

 is to be taken b times ; that is, we must 

 take ab. 



3. -f-aX b = ab : for a quantity is 

 said to be multiplied by a negative num- 

 ber b, if it be subtracted b times ; and 

 a subtracted b times is ab. 



4. X b=x*\-ab. Here a is to 

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

 be subtracted ; but subtracting a b is 

 the same as adding + ab ; therefore we 

 have to add + ab. 



The 2d and 4^ cases may be thus prov- 

 ed; a a=o, multiply both sides by b, 

 and ab together with aX b must be equal 

 to bx> or nothing; therefore, a mul- 

 tiplied by b must give ab, a quantity 

 which when added to ab makes the sum 

 nothing. 



Again, a a=o / multiply both sides 

 by b, then a b together with ax 

 b must be =o,- therefore a X b 



If the quantities to be multiplied have 

 co-efficients, these must be multiplied to- 

 O 



