Cor. 1. Hence if the sum and difference of any two numbers be given, we can readily find each of the numbers; thus, if s be equal to the sum of two numbers, and d equal to the dif s+d ference; then the general expression for the first, is 2 Whatever may be the numeral values that we assign to s and d, or whatever values these letters must represent in a particular question, we have but to substitute them in the above expressions, in order to ascertain the numbers required: For example, Given the sum of two numbers equal to 36, and the difference equal to 8: s+d Then, by substituting 36 for s, and 8 for d, in and 2 28 2 14. So that, 22 and 14 are the numbers required. Cor. 2. Also, if it were required to divide the number s into two such parts, that the first will exceed the second by d. It appears evident, that the general expression for the first part is s+d s-d and for the second ;s and d representing any numbers whatever. 2 s+d 103. The general expression may be found after the 2 manner of Garnier. Thus, let a represent the first part; then according to the enunciation of the question, x-d will be the second; and, as any quantity is equal to the sum of all its parts, we have therefore, x+x-d=s, or 2x-d-s. This equality will not be altered, by adding the number d to each member, and then it becomes, 2x-d+d=s+d, or 2x=s+d; s+d 2 dividing each member by 2, we have the equality, x= ; in which we read that the number sought is equal to half the sum of the two numbers s and d; thus the relation between the unknown and known numbers remaining the same, the question is resolved in general for all numbers s and d. 104. We have not here the numerical value of the unknown quantity; but the system of operations that is to be performed upon the given quantities; in order to deduce from them, according to the conditions of the problem, the value of the quantity sought; and the expression that indicates these operations, is called a formula. It is thus, for example, that if we denote by a the tens of a number, and the units by b, we have this constant composition of a square, or this formula, a2+2ab+b2; this algebraic expression is a brief enunciation of the rules to be pursued in order to pass from a number to its square. 105. From whence, we infer that, if a number be divided into any two parts, the square of the number is equal to the square of the two parts, together with twice the product of those parts. Which may be demonstrated thus; let the number n be divided into any two parts a and b ; Then n=a+b, and n=a+b; by Multiplication, n2=a2+2ab+b2 (Art. 50). 106. If the sum and difference of any two numbers or quantities be multiplied together, their product gives the difference of their squares, observing to take with the sign that of the two squares whose root is subtracted. - Let м and N represent any two quantities, or polynomials whatever, of which м is the greater; then (M+N)X(MN) is equal to m2-N2; for the operation stands thus; (MN)X(M-N)=M2+MN 107. When we put м=a3, and N=b3; then, (a3+b3)×(a3—b3)=a°—b® ; (See Ex. 9. page 30). Where a is the square of a3, and 6 that of b3, and this last square is subtracted from the first. Reciprocally, the difference of two squares m2-N2, can be put under the form (MN)X(MN). 1 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 =x+a; =x2+ax+a2; =x3+ax2+a2x+a3; α -= x2+ax3+a2x2+a3x+aa ; a X- α =x3+ax1+a2x3+a3x2+a*x+a3 ; We may conclude that in general, m-am is divisible by x--a, m being an entire positive number; that is, 109. The difference 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 even number. For since -— x3—ax2+a3x — a3 ; &c. &c. Hence we may conclude that, in general, x2m-a3m x+a 2m⋅ +a2m-2x—a2ni−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. 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 a 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 x3 — x2, oг x2(xm—~—1) . . . . . . (7), is divisible by x-1, whether m-n be even or odd; and that xTM — x2, oí x2(xm-n—1) . . . . . . (8), is divisible by x+1, where m—n is an even number; as also that xmx", or xn(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 a b3c, for example, which comprehends six simple factors, is of the sixth degree; this, ab3c 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 factors. Thus, in Ex. 1, page 29, 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 31, 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. 55 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 may be a whole It may be observed, that the fraction cd 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 when we divide both with a, we shall have a=b ×v... ... (1). sides by b which has no factor common a Thus v will represent the value of the fraction, or the quo tient of the division of a by b. |