may be done, since it is only asserting the same proposition in an inverted order; = . (111) (a+b)c-be ac, subtracting bc from both sides; but, as a and b may be any quantities whatever, a+b may be any whatever, and we may substitute m for a+b, n for b, and, consequently, m―n for (a + b) — b = a; Scholium. The last form, resulting from the hypothesis (5.) that m is greater than n [(a+b)>b], must also be adopted when m is less than n; otherwise general symbols for the representation of quantities would have to be abandoned altogether, as it will frequently be impossible, as well as generally inconvenient, to distinguish between m and n, whether m be greater or less than n. And in order to thus extend the application of the form, or to make it general, we have only to interpret the expressions mc (m-n)c, both when m is greater and less than n : 1°, when m>n, we have mc> nc and, consequently, mc plus; but 20, when m < n, we find mc <nc, and mc both become minus. - nc, nc, m n, both nc, m n, It is easy now to extend the operation to any number of terms, whether plus or minus. Thus, ax + bxcx+ dx + ... = (a+b)x-cx + dx +... =(a+b-c)x+ dx+... =(a+b-c+d)x + ... =(a+b-c+d+...)x. So the first part of the proposition is proved. 2o. Dividing both sides (112) of the last equation by x, we have (ax + bx − cx + dx + ... ) : x = a+b-c+d+..... = · (ax): x+(bx): x − (cx): x + (dx) : x + ..., and the proposition is proved. Cor. 1. A Polynomial, or algebraical expression consisting (55) of many terms, is multiplied or divided by multiplying or dividing its terms. Cor. 2. One polynomial is multiplied into another by mul- (56) tiplying all the terms of the one into all the terms of the other. Thus, (a+b+c+ ...) (a2 +b2 + C2 + ...) = (a+b+c+ .....) a2 +(a+b+c+ ...) b 2 + ( a + b + c + ...) c2 + ... = aa2+ba2+ca2 + ... + ab2+bb 2 + cb2 + ... + ac2 + bc 2 +cc2 +..... (56). 2 Cor. 3. The number of terms in a polynomial product, is (5) equal to the product of the numbers denoting the terms in the constituent polynomial factors. Thus, if P., P¿, P., ... [m], denote polynomials of a, b, c, terms, the polynomials being m in number, we have and generally Pa P b = Pa ab a polynomial of a times b terms; P.. P P = Pab X Pe= Pabe a ... If we make the number of terms the same, a for instance, in all the Thus, the number of of the binomial x+y, Cor. 4. The number of terms in the mth power of a polynomial of a terms, is equal to the mth power of a. terms in the expansion of the sixth power [(x + y)] will be found to be 26 = 64. Def. 6. Operations like those preceding, which may be performed upon the whole of a polynomial at once, or upon its parts separately, are denominated linear. Cor. 5. The compounds of the above linear operations, (510) are themselves linear. Thus, if we multiply any polynomial, x+y+z+..., by any quantity a, and then divide by b, we find [a (x+y+z+...)] : b = (ax): b+ (ay): b+(az): b+.... PROPOSITION VI. A product made up by the multiplication of additive (6) quantities, is itself additive; and is changed in sign by changing the sign of any one of its factors. This proposition will become evident by comparing (53) and its extension in (54) with the first part of (4); for, observing that m -n may represent any quantity either plus or minus, by making m greater or less than n by that quantity, and that m are both plus when m is greater than n, [m>n],and both minus when m< n, (m − n) c = mc = -= + n, mcnc nc, becomes + (m − n) • + c + (mc · nc), or + • + = + when m>n, and • +c (mc - nc), or = - when m<n; but the order of the factors may be changed at pleasure, c (mn) = (m − n) c, .. + •+=−: from all which it follows that changing the sign — of a factor changes the sign of the product into which it enters. Cor. 1. An even number of minus factors gives a plus (62) product-an odd number, a minus product. Cor. 2. An even power of a plus or minus quantity is (63) plus (62). Thus (+a)2= + a2, (− a)2 = (±a)2n=+a2n. = a • — a = + a2, Cor. 3. An even root of a plus quantity is either plus or (64) minus [+], (63). Thus 2+ a2 =±a. Cor. 4. An even root of a minus quantity is imaginary, (65) that is, impossible; for (64) it can be neither plus nor minus. 2n Thus, in a±r, r is imaginary; for, raising to the (2n)th pow er, we have Cor. 5. An (62). Thus, 2n 2n -=+impossible. a= |— a) 2n = (±r)2", = + p2n, SQUARE OF POLYNOMIALS, 31 Cor. 6. An odd root of a minus quantity is minus (66). (67) Cor. 7. In products and quotients, like signs [+, +, or-,-] (6.) produce plus [+] and unlike signs [+, -, or -, +,], minus [-]. For products this has already been shown (62), and, to establish the same for quotients, let q be the quotient arising from dividing any quantity D by any other d. or whence (112) or (4) D:d=q, (D: d) • d = qd, multiplying both sides by d, = D = qd ; Cor. 8. In products and quotients the signs +, -, are (6,) relatively free (6.). Thus, +• ++: PROPOSITION VII. = The square of a polynomial is equal to the sum of the (7) squares of the terms and their double products taken two and two. For in (56) making a2 = a, b2 = b, c 2 = C, and arranging, we (a+b+c+...)2 = a2 +b2 + c2 +...+2ab+2ac+... have +2bc+... 2 (7) (8) Cor. 1. The square of a binomial is equal to the sum of (8) the squares of its terms increased by their double product. For in (7) making all the letters nothing except a and b, there results (a+b)2 = a2+b2+2ab = a2+2ab+b2 = a2 + (2a + b) b. · Cor. 2. The square of a residual is equal to the sum of the squares of the terms diminished by their double product. For changing the sign of b in (8) we have The product of the sum and difference of two quantities (10) is equal to the difference of their squares, and vice versa, the difference of the squares of two quantities is equal to the product of their sum and difference. For in (5) making all the letters nothing except a, b, a, b, and changing a, into a, b, into b, we get Q. What relation does (56), bear to (7)? (7) to (8)? (8) to (9)? (56) to (10)? Which is the more general (56) or (7) ? Scholium. Forms (7), (8), (9) and (10), are important for their applications, and remarkable for illustrating the facility with which general truths are discovered by the employment of algebraical language. PROPOSITION IX. A polynomial may be freed from the minus sign, or on (11) the contrary subjected to its influence, by changing the signs of all its terms. Every polynomial, as 3a5b+c+ 2d - e, may be, so far as the signs are concerned, reduced to the form + B-C, representing the sum of the plus terms, as 3a+c+2d, by B, and the sum of the minus terms, as 56 e, by C. Therefore all cases of subtraction will be comprised in this general form, A-(+B-C), where it is required to take the polynomial (+B-C) from any quantity A. Now if from A we subtract B, the sum of the plus terms, we have A-B, by which all the terms in B originally + become ; but, in taking the whole of B from A, we have diminished A by a part of B, namely C, which ought not to have been subtracted, since the true subtrahend, BC, is only that part of B remaining after the diminution of B by C; therefore, A- B being too small by C, the true remainder becomes A- B+C, on the addition of C-and all the terms embraced in (+B-C) change their signs in (- B+C), also or or A-(+B-C) = A-B+ C, -B+C=-(+B-C.) Q. E. D. Remark. The student should be familiar with the resolution of polynomials into factors, not only by the addition and subtraction of coefficients, but by the employment of the theorems under (7), (8), (9), (10). |