x2m-a2m x+a ¡2m−1..(2). = x2m—1—α x 2m-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-αx+x2; · = x2¬ax3+a2x2 —a3 x+a1 ; Hence, we may conclude that, in general, x2m+1+a2m+1 x+a = x2m -ax 2m-1+ 2m-1 —a2m¬1x+a2m.. (3). 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 formulæ, be taken =1, and x being always considered greater than a; they will stand as follows: 113. And if any two unequal powers of the same root be taken, it is plain, from what is here shown, that ¿1—x", or x2(xm--n — 1 ) (7), is divisible by x-1, whether m-n be even or odd; and that is divisible by x+1, when m-n is an even number; as also that ..... x2+x”, or x2(xm--n + 1) . . (9), 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 b2c 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 ao edmn cd It may be observed that the fraction ab cd. may be a 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 α Thus 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=mb xv..... (3); dividing both by mb, we shall have 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, or the denominator divided by it. For, if we divide by b, the two members of the equality (3), we obtain the following, The equality (1) may also be put under the form m whence we derive, dividing each side by b, 120. If a fraction is to be divided by m, it is the same whether the numerator be divided by m, or the denominator divided by it. For, from the equality (1), we deduce these dividing the first by b and the second by mb, in order It is to be observed that in, the numerator is a m and the denominator.b, and that we employ the greater line for separating the numerator from the denominator. 121. If two fractions have a common denominator, their sum will be equal to the sum of their numerators divided by the common denominator.. For, let now the two equalities be corresponding to the fractions |