DIVISION.

93. To divide one quantity by another is to determine how often the latter is contained in the former, or what quantity multiplied by the latter will produce the former.

Thus to divide ab by a is to determine how often a must be taken to make up ab, that is, what quantity multiplied by a will give ab; which we know is b.

From this consideration are derived all the rules for the division of algebraical quantities.

The words Dividend, Divisor, and Quotient, have the same meaning here as in common Arithmetic.

94. If the divisor and dividend be affected with like signs, the sign of the quotient is +; but if their signs be unlike, the sign of the quotient is -.

Thus,

If ab be divided by a, the quotient is +b; because - a x+b gives - ab; and a similar proof may be given in the other

cases.

95. To divide one SIMPLE quantity by another.

RULE. In the division of simple quantities, if the divisor be found as a factor in the dividend, the other part of the dividend, with the sign determined by the last rule, is the quotient.

This is obvious, since Quotient× Divisor = Dividend in all cases. Thus -76÷b=-7; -ax÷-a=x; 14ab÷7b=2a.7b÷7b=2a. Also abc÷ab = c; because ab multiplied by e gives abc. If we first divide by a, and then by b, the result will be the same; for abc÷a=bc, and bc÷b=c, as before.

COR. If any POWER of a quantity be divided by any other POWER of the same quantity, the quotient is the same quantity with an index which is found by taking the index of the divisor from the index of the dividend.

=

Thus a'aa'xa÷a" (Art. 91)=a', (Art. 84); and generally, if m and ʼn be positive integers, and m>n, a"÷a"=a"-"×a"÷a"=a"~".

Similarly 6ab÷3a*b3=2ab3×3a2b3÷3a2b3=2a2b3; and (a"b")÷(a2b3)=: aTM-ol"-¶ ̧ a2l¶÷aoLo = aTM-ol"¬¶ ̧

a

96. LEMMA. To shew that a÷b is equal to the fraction.

a

According to the definition of a 'fraction' the unit is divided into b equal parts, and a of them are taken, to make the quantity represented by ī' Now, each of these parts is clearly the 6th part of the unit, and .. is equal to a times the 6th part of 1,=x, suppose. Multiplying these equals by b, (Art. 81), bxa times the 6th part of 1, or axb times the 6th part of 1=bx; but b times the 6th part of 1 is clearly 1,.. a=ba. Now let a+b=y, then by Definition (Art. 93), a=by, .. bx=by, or x=y, (Art. 82); that is a÷b.

a

b=a÷b.

97. In dividing one simple quantity by another, if only a part of the product which forms the divisor be contained in the dividend, the division must be represented by a fraction according to the direction in the last Art., and the factors which are common to the divisor and dividend expunged.

Thus 15a3b'c÷- 3a2bx =

For, 1st, divide by

Sab, and the quotient is -5abc; this quantity is still to be divided by x (Art. 95), and as a is not contained in it, the division can only be represented in the usual way;

98. To divide a quantity of two or more terms by a simple quantity.

RULE. If the dividend consist of several terms, and the divisor be a simple quantity, every term of the dividend must be separately divided by it.

Thus to divide a3x2 - 5abx3+ 6ax1 by ax3,

ax ax2 ax2

a b с 1 1 1
+ +
+

Also (a+b+c)÷abc=abc abc abcbc

ac ab

99. To divide one quantity by another when the DIVISOR consists of two or more terms.

RULE. When the divisor consists of several terms, arrange both the divisor and dividend according to the powers of some one letter* contained in them, (that is, beginning with the highest

The operation will be shortest when that letter is chosen whose highest power in the dividend comes nearest to the highest power of the same letter in the divisor; and the same arrangement according to the powers of that letter must be kept up throughout the whole operation.-ED.

power and going regularly down to the lowest, or vice versâ,) then find how often the first term of the divisor is contained in the first term of the dividend, and write down this quantity for the first term in the quotient; multiply the whole divisor by it, subtract the product from the dividend and bring down to the remainder as many other terms of the dividend, as the case may require, and repeat the operation till all the terms are brought down.

Ex. 1. If a2 - 2ab+b2 be divided by ab, the operation will be as follows:

The reason of this and the foregoing rule is, that as the whole dividend is made up of all its parts, the divisor is contained in the whole as often as it is contained in all the parts. In the preceding operation we inquire first, how often a is contained in a2, which gives a for the first term of the quotient; then multiplying the whole divisor by it, we have a2- ab to be subtracted from the dividend, and the remainder is – ab + b2, with which we are to proceed as before. The whole quantity a2 – 2ab+b2 is in reality divided into two parts by the process, each of which is divided by a b; therefore the true quotient is obtained.

Ex. 2.

a+b)ac + ad + bc + bd (c + d

ac + bc

ad + bd

From this example it appears that y-1 is divisible by y-1 without remainder, the quotient being y+y+1. It may be shewn in the same manner that -a is divisible by x-a, the quotient being a2+ax+a2; and that 3+a3 is divisible by x+a, the quotient being r2-ax+a'. These results are worth remembering.

Ex. 5. To divide 4ab3+51a2b3+10a-48a b-156' by 4ab-5a+3b3. First arrange the terms of both dividend and divisor according to the powers of a, beginning with the highest.

-5a+ab+36) 10a-48ab+51a b'+4ab3-156' (-2a+8ab-5b*

This division will obviously terminate without remainder for any proposed integral and positive value of m, when the quotient has reached tom terms, the last term being y-1. Hence we have,

(x3—y3)÷(x−y)=x2+xy+y2,

(x'—y1)÷(x−y)=x3+x3y+xy2+y3,

(x3-y®)÷(x−y)=x*+x3y+x'y2+xy3+yʻ; and so on.

Ex. 7. x-ax3-px2 + qx − r ( x2 + (a− p)x + a2 − pa + q

x3- ax2

(ap)x2 + qx

(ap)x2 - (a2 - pa)x

+(a2-pa + q)x-r

(a2 - pa + q)x− (a3 — pa2 + qa)

Remainder a3 - pa2 + qa-r

[Exercises E.] See NOTE 1.

7

TRANSFORMATION OF FRACTIONS TO OTHERS OF EQUAL VALUE.

100. If the signs of all the terms both in the numerator and denominator of a fraction be changed, its value will not be altered.

101. If the numerator and denominator of a fraction be both multiplied, or both divided, by the same quantity, its value is not altered.

For in the unit is divided into b equal parts, and a of them are taken,

b

a

in Scr

bc

......bc......... c times as many as before,

and a of them taken; therefore in the former case each part is c times as great as in the latter. But if c times as many of the smaller as of the greater parts be taken it is obvious that the result in either case will be the same- —that is, that ac parts of the latter are equal to a parts of the

Hence a fraction is reduced to its lowest terms by dividing both the numerator and denominator by the greatest quantity that measures them both.

This quantity is called the Greatest Common Measure, or Highest Common Divisor, of the numerator and denominator.

102. A fraction which has either its numerator or denominator a simple algebraical quantity is easily reduced to lowest terms; for the greatest common divisor of the numerator and denominator is at once found