Let and N represent any two quantities, or polynomials whatever, of which м is the greater; then (M+N)X(M-N) is equal to м2-N2; for the operation stands thus;

(M+N)×(M—N)=M2+MN

-MN-N3

}

107. When we put м=a3, and N=63; then, (a3 +b3) × (a3 —b3)—a®—b° ; (See Ex. 9. page 41). Where a is the square of a3, and bo that of b3, and this last square is subtracted from the first. Reciprocally, the difference of two squares м2 —N2, can be put under the form (M+N) × (m—n).

M2.

This result is a formula that should be remembered.

108. The difference of any two equal powers of different quantities is always divisible by the difference of their roots, whether the exponent of the power be even or odd. For since

x5-α5

6

3C -aв

X- -a

&c.

= x2+ax2+a2x2 + a3x+α^ ;

5

3

· = x2+ax1+a2x3 + a3 x2 + a1x+a3;

&c.

We may conclude that in general, m-am is divisible by x-a, m being an entire positive number, that is,

cm-1+axm-2+ +am-2x+am--1

109. The difference of any two equal powers of afferent quantities, is also divisible by the sum of their roots, when the exponent of the power is in even number. For since

x2m-a2 x+a

2m

2m-2
=x2m-1 ax 2+..+a2m-2x

-a2m-1

(2). 110. And the sum of any two equal powers of different quantities, is also divisible by the sum of their roots, when the exponent of the power is an odd number. For since

x+a

= x2-ax+x2;

3

= x2 — ax3 + a3 x2 —a3x+ã1 ;

Hence, we may conclude that, in general,

x2m+1+a2m+1

x+a

=x2m

111. In the formulæ (1), (2), (3), as well as in all others of a similar kind, it is to be observed, that if m be any whole number whatever, 2m will always be an even number, and 2m+1 an odd number; so that, 2m is a general formula for even numbers, and 2m+1 for odd numbers.

112. Also, if a in each of the above formula, be taken 1, and x being always considered greater than a; they will stand as follows:

=x3-1+xm−2+xm-3+....+x+1... (4),

m

x

-1

113. And if any two unequal powers of the same root be taken, it is plain, from what is here shown, that

m--n
x-x", or x^(xm 1) . . . . (7),

is divisible by x-1, whether m-n be even or odd; and that

x-x", or x(xm--n — 1) . . . . . ...(8), is divisible by x+1, when m-n is an even number; as also that

x+x", or x(xm-n+1)...... (9),

x2+x”,

is divisible by x+1, when m-n is an odd number. 114. It is very proper to remark, that the number of all the factors, both equal and unequal, which enter in the formation of any product whatever, is called the degree of that product: The product ab3c, for example, which comprehends six simple factors, is of the sixth degree; this, a ̃bae is of the tenth degree; and so on.

Also, that if all the terms of a polynomial, or compound quantity, be of the same degree, it is said to be homogeneous. And, it is evident from the rules established in Multiplication, that if two polynomials be homogeneous; their product will be also homogeneous; and of the degree marked by the sum of the numbers which designate the degree of those fac

tors.

Thus, in Ex. 1, page 39, the multiplicand is of the fourth degree, the multiplier of the third, and the product of the degree 4+3, or of the seventh degree.

In Ex. 12, page 42, the multiplicand is of the third degree, the multiplier of the third, and the product of the degree 3+3, or of the sixth degree.

Hence, we can readily discover, by inspection only, the errors of a product, which might be committed by forgetting some one of the factors in the partial multiplications.

76

CHAPTER II.

ON

ALGEBRAIC FRACTIONS.

115. We have seen in the division of two simple quantities (Art. 84), that when certain letters, factors in the divisor, are not common to the dividend, and reciprocally, the division can only be indicated, and then the quotient is represented by a fraction whose numerator is the product of all the letters of the dividend, not common to the divisor, and denominator, all those letters of the divisor, not common to the dividend.

Let, for example, abmn be divided by cdmn; then,

abmn ab

ab

It may be observed that the fraction may be a

cd

whole number for certain numeral values of the. letters a, b, c, and d; thus, if we had a=4, b=6, c=2, d=3; but that, generally speaking, it will be a numerical fraction which can be reduced to a more simple expression.

§ I. Theory of Algebraic Fractions.

116. It is evident (Art. 103), that if we perform the same operation on each of the two members of an equality, that is, upon two equivalent quantities or numbers, the results shall always be equal.

It is by passing thus from the fractional notation to the algorithm of equality, that the process to be pursued in the researches of properties and rules, becomes simple and uniform.

117. Let therefore the equality, be

a=bxv

(1′),

when we divide both sides by b which has no factor common with a, we shall have

Thus v will represent the value of the fraction or the quotient of the division of a by b.

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

For, if we multiply by m the two members of the equality (1), we will have these equivalent results, ma=mbXv. . . . . . (3) ; dividing both by mb, we shall have

mb

b

m being any whole or fractional number whatever. 119. If a fraction is to be multiplied by m, it is the same whether the numerator be multiplied by it, ar the denominator divided by it.

For, if we divide by b, the two members of the equality (3), we obtain the following,