These also may be similarly denoted:

a3+b3
a+b

= a2-ab+b2; and

a3 - b3
a-b

= a+ab+b2.

And in the same manner, if a3+3a2b+3ab2+b3 be divided by a+b, and a3-3a2b+3ab2 - b3 by a-b, it will be found that the quotients are a3+2ab+b3 and a2-2ab+b2 respectively, and that

The quotients arising from the following dividends and divisors are important.

1. a"-b" is exactly divisible by a-b when the n is any positive

integer odd or even.

2. a+b" is exactly divisible by a+b when n is an odd integer.
3. a"-b" is exactly divisible by a+b when n is an even integer.
4. a+b" is in no case exactly divisible by a-b, whether n be an
odd or even integer.*

To shew that a"-b" is divisible by a-b, when n is odd or even.

a—b)a" — br (an−1 +an-2b+an-sb2+.

a"-b"-1

+an-1b-br

+an-1b-a-2b2

It is obvious that at each step of the division, the indices of the powers of a in the remainders are successively diminished by unity, and after n steps there will be no power of a in the remainder.

Let Q denote the quotient and R the remainder,

then a"-b" (a−b) Q+R,

and since the remainder R does not contain a, it will not be altered on giving any value whatever to a.

Let a=b, then b′′ — b′′ = (b—b) Q+R, or R=0,

that is, the remainder is 0 after n terms of the quotient have been obtained, whether n be an odd or even number, and the quotient with the dividend and divisor may be denoted thus:

an-fr

a-b

:

If n=2, 3, 4, 5, &c., any integral numbers,

as-b3
a--b
-=a1+a3b+a2b2 +ab3 +b1, and so on.

=a2+ab+b2,

at-b4
a-b

=a3+a2b+ab2+b3,

If dividends whose coefficients are rational of the form x3±px+q, x3±px2±qx±r, x1±px3±qx2±rx±8, x3±pa1±qx3±rx2±sx±t, &c., are divided respectively by xa; after two, three, four, five, &c., terms of the quotient have been determined, the remainders will be found to be of the same form as the dividends with a in the place of x, in some cases with the same signs as the dividend, in others with different signs.

If x2+px2+qx+r be divided by x-a,*

The coefficients of the dividend are 1, p, q, r.

The coefficients of the quotient are 1, a+p, a2+ap+q,

And the remainder is a3+pa2+qa+r.

This remainder is obviously the same expression as the dividend, having a in the place of x.

Now if a be actually substituted for x in the dividend, this assumes that xa, or x-a=0; hence, when a is substituted for x in the dividend, the remainder will be equal to 0, and x3+px2+qx+r will be exactly divisible by x-a.

When any given expression with numerical coefficients is or is not divisible by x+a, the quotient and the remainder can be found by substituting the numerical values for the general coefficients of the dividend and divisor.†

* The following is the process of the division exhibited, and the proof of its correctness :

x-α) x2+px2+qx+r(x2+(a+p)x+(a2+pa+q)

x3-ax2

The expression 23— 6x2+11x−6 is exactly divisible by x-3, and gives '23 −3x+2: for the quotient.

Is x3+7x2+9x+12 divisible by x+3?

The coefficients of the dividend are 1, 7, 9, 12.

The coefficients of the quotient are 1, 4, 39.

And the remainder is 33+ 7·32+9·3+12=129.

The expression x+7x+9x+12 when divided by x+3 gives the quotient x2+10x+39, with a remainder 129.

Also the conditions may be readily found when any rational expression of these forms is divisible by x+1 and by x—

—1.

If x2+px2+qx+r is divisible by x+1, +1 must be equal to 0, and 1 by substituting this value for x, the dividend becomes x = (-1)+p(-1)+(-1)+r, which must also be equal to 0, or -1+p-q+r=0, or p+r=q+1, that is: the expression x2+px2+qx+r is divisible by x+1, if the sums of the alternate coefficients of the dividend are equal.

And if +p+qx+r is divisible by x-1, then x-1 must be equal to 0, and x=1, and substituting this value of x, 1+p+q+r must be equal to 0, or the sum of coefficients of the terms of the dividend must be equal to 0.

14. The products and quotients of concrete quantities admit of different interpretations, according to the nature of the magnitudes denoted by the symbols. In their application to geometrical magnitudes, they include lines, surfaces, and volumes, as well as angles and curves. The straight line has one dimension, length only; the surface has two dimensions, length and breadth; and the volume has three, length, breadth, and thickness.

The unit of length is arbitrary, and may be any straight line of definite and known length.

The square is the figure assumed as the measure of surfaces, and the unit of area is assumed to be a square, any side of which is an unit in length. The cube is the figure assumed as the measure of volumes, and the unit of volume is a cube, any edge of which is an unit in length.

To find the measure of any line, surface, and volume, is to determine the number of times any length, surface, and volume contains the unit of length, surface, and volume respectively.

If a, b denote the number of lineal units in two adjacent sides of a rectangular parallelogram or rectangle, and ▲ the number of square units in the area, it may be shown that A = ab*; that is, the number of square units in the area of a rectangle is denoted by the product of the two numbers which express the number of lineal units in the two adjacent sides of the rectangle. Or, in other words,

If a number representing a line be multiplied by another representing a line, the product is a number representing an area.

And since 4 = ab, it follows that 4b, a number denoting & lineal

units.

A

a

Hence. If a number representing an area be divided by another representing a line, the quotient is a number representing a line.

* The reader is referred for the proof of the expressions A=ab, and Vabc, to the notes on the Second and Eleventh Books of the author's edition of Euclid's Elements of Geometry, with geometrical exercises selected from the Cambridge Examination Papers.

Again, if a, b, c denote the number of lineal units in the three adjacent edges of a rectangular parallelopiped, and V denote the number of cubic units in the volume, it may be shown that the product of the numbers a, b, and c is a correct representation of the number of cubic units in the volume of the parallelopiped, and that Vabc. Or in words,

The number of cubic units contained in the volume of a rectangular parallelopiped, is expressed by the number of cubic units contained in the product of the numbers of lineal units in any three adjacent edges of the parallelopiped.

abc A ab

And since V = abc, and ▲ = ab, then =c, a line of c units. Or, if a number representing a volume be divided by a number representing an area, the quotient is a number representing a line. Next, V = abe and = ab, an area of ab square units.

с

Or, if a number representing a volume be divided by another representing a line, the quotient will be a number representing an area. It will also be obvious, that if a number representing a line, an area, or a volume be multiplied by any abstract number, the products will each be a multiple or submultiple of the line, the area, or the volume respectively.

And in the same manner, if a line, an area, or a volume be divided by an abstract number, the quotient will denote some part of the line, the area, or the volume respectively.

And if a number representing a line be divided by another representing a line, the quotient will indicate the number of times the one contains the other, and is therefore an abstract number.

In the same

manner, a number representing an area divided by another representing an area, as also a number representing a volume divided by another representing a volume, will denote abstract numbers in the quotients.

In a similar manner the products and quotients of other concrete quantities may be interpreted; as, for instance, if a body move uniformly over a distance of 8 units in t units of time, then will denote t

8

the distance which the body has moved over in one unit of time, and is defined to be the velocity of the body. If v be assumed to denote this distance or velocity, then v, and s tv, or, the distance moved over

8

=

t

=

with an uniform velocity, is equal to the product of the numbers which denote the velocity and the time.

EXERCISES.

I.

1. What operation bears the same relation to multiplication that multiplication bears to addition?

2. Explain what is meant by a reverse operation in Algebra, and give an example.

3. Explain the meaning of the terms factor and dimension as applied. to Algebraical expressions. Give examples.

4. What is meant by the rule of signs in the multiplication of Algebraical quantities? State whether you regard the rule as conventional or demonstrative, giving your reason in the former case, or a demonstration in the latter.

5. Define the meaning of a"; shew that a".a" = am+", and and explain the notations ao, a ̄”.

am

an

6. Explain and interpret the expressions ab, and abc, when a, b, c, represent straight lines.

7. Find the products of 5abx and 4cdy; mab and -nc; -12a-bc3 and 5ab'd3; -12a be3 and -8ab'd3 respectively.

8. Divide 20abcdxy by 4cdy; -mnabc by mab; —60a3b3c3ď3 by -12a2bc3; +96a3b3c3d3 by -8ab3ds, respectively.

9. Multiply 9a+2x by 3a; 4a-8b by -3ab; x2-ax+a2 by −ax ;. 3a2-4ab-56 by -6a2bc2; max-nay+paz by axyz; —a2x2+b2y2 —c2x2 by xyz2, respectively.

10. Divide 27a2+6ax by 3a; -12ab+24ab2 by-3ab; -ax3+a2x2—a3x by-ax; 18abc2+24a3b3c2+30a3b*c2 by-6a2b3c2; ma2x2yz-na2xy3z+pa2xyz2 by axyz; —a2x1y°z2+b2x2y1za—c2x2y11 by x2y2x2, respectively.

11. Shew the truth of, and express in words at length, the identical expressions:

1. a(a+b)+b(a+b).

2. a(a-b)-ba—b).

3. (a+b)3+(a−b)2 = 2(a2+b2). 4. (a+b)-(a—b)=4ab. 12. In the first ten propositions of the second book of Euclid's Elements, if "the rectangle contained by two lines" be exchanged for "the product of two numbers," and "the square on a line" for the square of a number:" shew that the propositions so changed are true for all numbers whatever.

13. In the multiplication and division of Algebraical polynomials, what preliminary steps are necessary in order to avoid confusion in performing these operations?

II.

Find the following products, and prove the correctness of them by reversing the operation in each example:

1. Of 2x+3y and 2x+3y; 2x-3y and 2x+3y; 2x-3y and 2x-3y.