MATHEMATICS -ALGBBRA. 



VTIOX. 



EXAMfLES FOR IXKBCISE. 

 2. 6 V 4 



-Sy 



In tli tollowing examples, the rinculnm or braelcla 

 may be removed, aud the simple factor (the multiplier) 

 placed under the leading tonn of the compound one, as 

 in OR- preceding instances ; but in most cases you will bo 

 able to write down the result of the multiplication at once, 

 vitliout resorting to thin arrangement. 

 G. (2a%r BrjrtSaz^ 



(2axy 3xz-j-2*)X 4azy 

 5 + 3i v 5 2y*;) X Gm*c*y 

 So;? 5&y") X 5ai'y 

 i 3z) 12c*xV 

 2cz*)X 12a*6c 



|4ay* (2M:' yj 2)3aV 



X 



14. {faV (31>V + 2a V 4ay 3 ) } X 

 CASE III. fP&* muWijJicamJ and multiplier are both 

 compound quantities. 



ROLE 1. Multiply all the terms of the multiplicand 

 by each term of the multiplier, proceeding with each as 

 in the last case. 



2. Collect together tho several products that are like, 

 as in addition, and to the sums of these unite, by their 

 proper signs, the other products ; and the complete pro- 

 duct will be obtained. 



L Multiply z -J- y by x y. 



x + y 

 z y 



-y ! 



2 Multiply x + y by x -f- y. 



3. Multiply x y by x y. 



- V) =*'- 



We hare a remark to make in reference to these three 

 example* ; you will do well to keep it in remembrance. 

 It u this : since x and y stand for any two numbers 

 whatever, wo learn that 



1. The sum of two numbers (x+y) multiplied by their 

 difference (z y) is the difference of tho squares of those 

 numbers (z* y*). 



2. The square of the sum (jr-f-y)of two numbers (x and y) 

 it equal to the sum < f their squares (z*+y a ), together 

 with twice their product (2xy). 



3. The square of the difference (x y) of two numbers 

 u equal to the sum of their squares (z*+y 2 ), diminished 

 by twice their product (2xy). 



EXAMPLE. Let 7 be one numlicr and 3 the other : 

 then, 1st, their turn, 10, multiplied by their difference, 



4, U 40; the tquant of tin- numbers are 49 and 9; so 

 that 7 3- (<+3) (7 3)-40. 2nd. (7+3)- 10* = 

 100, and 7 * +3* -49+9- 68 ; also twice 7x3, that is, 

 2x7X3-42; aud 68+42-100. 3rd. (7 3)*-4 = 16 ; 

 and 68 42-10. 



4. Multiply (x -j-y)* by x + y, that is, multiply outor 

 develop* 



x+y 



. 



6. Dttelope, or multiply out (x 4) (x + 3); (x + 5) 

 (x 7); (2x+3)(:U + 2)and(x + a)^ !,). 

 x 4 z + 5 



z + 3 z 7 



z*+5z 

 7x 35 



z 8 4x 



3x 12 



x 2 z 12 



2x + 3 

 3x+2 



z 2x 33 



4x+C 



x- -{- ax 



+ 06 



In the third of these results, the 13x is got by actually 

 adding the coefficients 9 and 4 : in the fourth example, 

 the corresponding corili. -u'uts of x, not being numbers, 

 cannot be actually added; but the addition may never- 

 theless bo indicated, as above : aud where, in the third 

 example, the middle term of the product was written 

 13z, that is, (9+4)x, the corresponding term here is 

 written (a-f-b)x; this is better than making tico terms of 

 the same quantity, and writing it thus, ax-f-bx. Here 

 the coefficients of x are o and 6; and, being letters, tla-y 

 are called literal cw//Vi-iVn/.s: the coefficient of x, in the 

 result above, is (a+l>). Whenever a common factor 

 enters several of the terms of a compound expression, 

 tliis common factor may always be written outside a 

 vincul urn, and whatever multiplies it, within; reped 

 of the same factor are thus avoided ; and in tho answers 

 to questions, or the final results of operations, this more 

 compact form of expression should be adopted. Thus, 

 such a result as o6x+4cx 3mx, would have au un- 

 finished appearance ; it should be changed into (a!-f !c 

 3m)x. This, you will see, is more easily compute! 

 than the former, when the letters are replaced by tho 

 numbers they represent. Suppose, fnr induce, a=12, 

 6 = 6, c = 5, and m=9; and that x=23 ; you will find 

 that the numerical Taluo of the expression is more 

 readily obtained from the second form thau from the 

 first. And we may as well tell you here, once for all, 

 that algebraists always take care that their i-i'sults are 

 presented in a form that will give the least trouble to tho 

 arithmetical computer. Tliia principle will in general be 

 observed in what follows. 



U c shall now work an example which suggests a 

 principle of some interest in arithmetic. 

 (0.) Multiply n + r by ri + r. 



n + r 



n'+r' 



nr'+rr' 



Now suppose a number to bo divided by any other 

 number, ay 9, and to leave a remainder, which we may 

 denote by r : then, if we express this numln i 

 the remainder, by n, the entire number will be denoted 

 by n -f- r. In like manner, a second number may bo 



