DIVISION. ] 



MATHEMATICS. ALGEBRA. 



463 



1. Divide 12a s x-y by Sax. Here 



4a-xy ; for the signs being unlike the sign of the quo- 

 tient is minus ; also 12 divided by 3 gives 4, so that 4 

 is the coefficient in the quotient ; and since, from look- 

 ing at the letters in dividend and divisor, we see that 

 two a's, one x, and a y, must be united to the divisor 

 to make up the letters in the dividend, these wanting 

 letters are those which the quotient must supply ; the 

 complete quotient is therefore 4a-xy. If the divisor 

 had been 3oxz. then the foregoing quotient must have 



3 * 

 been further divided by z; that is, 



for it is plain that to divide by 3oxz is the 



same as first to divide by 3oa; and afterwards by ^the 

 other factor z. In like manner, to divide by 3axy 2 is 

 the same as first to divide by 3oxiy, and then again to 



divide by the other factor y, 



All this is the same as in arithmetic ; thus, 



have to divide 48 by 30, we may proceed as follows : 



2 ' 4 \ ; for ~= 4, and this divided by the 



y 

 if we 



12.3~3' 



12 



: actor 3, gives 5* 



9 



3. 



* 



4. 



6. 



7. 



4a 

 \6a 1 x 3 ^y _ 



la* 

 xz 



2o 



9. 





2-/J _ 2,/z 

 3* 2 y 



NOTE. When the numerator of a fraction is mintu, 

 ami its denominator plus, it is matter of indifference 

 whether we put the minus before the numerator, or 

 before the entire fraction ; because, from the rule of 

 signs, a minus quantity is the result of the division in- 

 dicated. Thus, in tho last example it is indifferent 



2 J z _ 2 J z 

 whether we write ~^t~ or ""3^2 J~' T^e same is true 



when the denominator is minus, and tho numerator 

 plus ; for in division of one quantity by another, as 

 in multiplication, whichever of the quantities be 

 minus, provided only the other be plus, the result is 

 minus. Thus, the following all express the same thing : 



<j a a 



T = - r = T. For you see, that in either case, when 



a and 6 are interpreted, and the division performed, the 

 quotient is minus. For instance 



6 6 6 



Before proceeding to the following exercises, it will 

 be well for you to look again at precept 3 of the Rule, 

 from which you may yourself draw an inference of some 

 importance in the general theory of exponents. The 

 inference is, that when the same letter occurs in both 

 dividend and divisor, and that the exponent of it in the 

 former is greater than that in the Litter ; the quotient, 

 as far as this letter is concerned, is got by simply sub- 

 tracting the smaller exponent from the greater, and 

 placing the difference over the letter ; thus, in example 

 2, a* in the dividend, and a 1 in tho divisor, give a 3 in 

 the quotient, and x e in the dividend, and x 3 in the 

 divisor, give x 3 in tho quotient ; the y 2 in both is can- 

 celled. You thus see, that when the same letter or 

 quantity is concerned, division becomes tho 

 of exponents : multiplication, as you are aware, being 

 the addition of exponent*. You will no doubt think, 



The learner will perceive, that having indicated tho division by a 

 fraction, whether the fraction be algebraical or purely numerical, all 

 we have to do a to cancel all the /acton common to numerator and 



when the exponent in the divisor is greater than that in 

 the dividend, that this view of division must be aban- 

 doned ; but it is not so, as you will shortly see. 



EXAMPLES FOR EXERCISE. 



1. Divide 8ax-y* by 4oxy 2. Gl/x'y-z by Ibx'y 

 3. 12-r 3 y*z 5 by 3* 2 y 3 z' 4. IGo-r-y 3 by 4'jry* 

 5. 7aj 4 yz- by 5a-xyz 3 6. 8 *y/ b T 6* 3 y V- t 

 7. 24cy by 21c"y~z 8. 3Ga~j^"y by2X/ ( *"^ y 



9. 5a5x 2 -v^y by \Qol/~x\ y 



10. 3a = c 3 x* by 12< 



11. Imx'y 3 by 8m -x*y e 



12. 13j 2 y'zi by 2Gy* - 



CASE II. When the dividend is a compound quantity, 

 ami the divisor a simple quantity. 



RULE 1. Find the quotient of the divisor, and each 

 term of the dividend by the former rule. 



2. Connect the separate quotients together, by the 

 signs which belong to them, and the complete quotient 

 will be obtained. 



*- 2^-y- 



3. 



21aV + 7a* x* (42a*j 23). Here the com- 



pound term in the numerator is to be subtracted from 

 what precedes, therefore the signs of the subtractive quan- 

 tities are to be changed, and they are then to be added. J 

 tleiice the fraction is 



_ 3 ^ . ,_ 6fll + _4_ 

 " 



EXAMPLES FOB EXERCISE. 

 1. 9a j' 3a x' + 60*1 



3a* 

 2. 



3. 3aj (4xy 8ay')-f 12<uy 



Coxy 



4. 8xV 4(4og 2y).r 



2(3a*V*' 1) 



G. 



4ajry 

 ' (8 Sa'tx 4 ) 



CASE III. When both dividend and divisor are com- 

 pound quantities 



RULE 1. Arrange the terms of dividend and divisor so 

 that the exponents of the powers of some one letter in 

 both of them may appear in decreasing or increasing 

 order ; that is, if x, for instance, be the letter chosen to 

 govern the arrangement, place the terms in either of 

 these two ways : Let the term containing the highest 

 power of x be put first ; that containing the next power 

 immediately after, and so on ; or else let the term con- 

 taining the lowest power of x be put first, that containing 

 the next lower immediately after, and so on ; but do not 

 write the terms at random. 



2. The dividend and divisor, thus arranged, being 

 placed as dividend and divisor are placed in arithmetic, 

 divide the first term of the dividend by tho first term of 

 the divisor ; the result is the first term of the quotient. 



3. Then, as in arithmetic, multiply the ivhole divisor 

 by the part of the quotient thus found, and subtract the 

 product from the dividend. 



4. To the remainder annex another term, brought 

 down from tho dividend, or annex more of tho terms if 

 more are seen, from tho extent of the divisor, to be re- 

 quired ; the row of terms thus got is a new dividend, 

 with which, and the divisor, proceed as at first, and a 

 second quotient term will bo obtained. And in this wny 



denominator : tho result is the value of the fraction, reduced to its 

 simplest form. 



t See ante, p. 453. t See p. 455. 



