OPERATIONS.] When several quantities are multiplied together so as to constitute a product, each of them is called a factor of that product : thus a, b, and c are factors of the product abc ; also, a + x and b-x are factors of the product (a + x). (b-x). The products arising from the continual multiplication of the same quantity are called powers of that quantity, which is called the root. Thus act, aaa, aaact, &c., are powers of the root a. These powers are commonly ex pressed by placing above the root, towards the right hand, a figure, denoting how often the root is repeated. This figure serves to denominate the power, and is called its index or exponent. Thus, the quantity a being considered as the root, or as the first power of a, we have aa or a 2 for its second power, aaa or a 3 for its third power, aaaa or a 4 for its fourth power, and so on. The second and third powers of a quantity are generally called its square and cube. By considering the notation of powers, and the rules for multiplication, it appears that powers of the same root are multiplied by adding their exponents. Thus a x a 3 = a 4 , also x; 3 x x* = x 1 ; and in general a m x a" = a m+n . When the quantities to be multiplied appear under a symmetrical form, the operation of multiplying them may sometimes be shortened by detached coefficients, by symmetry, and by general considerations suggested by the particular examples under consideration. 13. Detached Coefficients. Ex. 1. Multiply a; 4 - Sx 3 + 2x 2 - *tx + 3 by x - 5x + 4. Here the powers of x occur in regular order, so that we need only write down the coefficients of the several terms during the operation, having it in our power to supply the x s whenever we require them ; we write, therefore, A L G E B E A 521 1-3+ 2- 1-5+ 4 7+ 3 1-3+ 2- 7+ 3 -5 + 15-10 + 35-15 + 4-12+ 8-28 + 12 1-8 + 21-29 + 46-43 + 12 The last line (for which the result might have been written down in full at once) is equivalent to a;G _ go; 5 + 2 1 a 4 - 2 9x 3 + 4 6.T 2 - 43:i< + 1 2 . When any terms are wanting, they may be supplied by zeros ; thus, Ex. 2. Multiply z 4 - 7ic 3 + x - 1 by a 3 - x + 2. Wo write 1 -7 1 +0 + + -14 1 2 - 1 - 1 - -14 -1 +1 + +2 -2 1 -7 + 04 - 1 + -f 1 . 7 - 2 x - z - x + te 4 - I5x 3 - the product required. 14. Symmetry. We may take advantage of symmetry by two considera tions either separately or combined. (1.) Symmetry of a Symbol. Ex. Find the sum of (a + b - 2c) 2 + (a + c - 2bY + (b + c-2a) 2 . Here a 2 occurs with 1 as a multiplier in the first square, with 1 as a multiplier in the second square, and with 4 as a multiplier in the third square, . . Ga 2 is part of the result ; ab occurs with 2 as a multiplier in the first square, with - 4 in the second, and Avith - 4 in the third . . Gab is part of the result. But a 2 , 5 2 , c 2 , are similarly circumstanced, as also ab, ac, be ; hence the whole result must be 6( 2 + 6 2 + c 2 ab ac -be). (2.) Symmetry of an Expression. Ex. Find the sum of (a + b + c) (x + y + z) + (a + b - c) First, the product of (a + b + c) by x + y + z is to be found by multiplying out term by term. It is ax + ay + az + bx + by + bz + ex + cy + cz. The product of (a + b-c) (x + y-t) is now simply tvritten down from the above, by changing the sign of every term which contains one only of the two quantities affected with a sign, i.e., in this case c and z. Lastly, the four products may be arranged below each other, the signs alone being written down ; thus, and the sum required is therefore kax + M>y + 4c2. 15. General Considerations. Ex. Find (a + b + c) 3 . By multiplying out we get (a + b) 3 = a 3 + 3a 2 & + ..... Now a, b, c are similarly involved in (a + b + c) 3 - . . I s and c 3 must appear along with a 3 , 3a~c, 3b 2 a, &c., along with 3a 2 &, and hence we can at once write down all the terms except that which contains abc. To obtain the co efficient of abc, we observe that if a, b, and c, are each equal to 1, (a + b + c) 3 is reduced to 3 3 or 27. In other words, there are 27 terms, if we consider 3a 2 i and every similar expression as three terms; and as the terms preceding abc are in this way found to be 21 in number, we require 6abc to make up the full number 27; (a + b + c) 3 = a 3 + 53 + c 3 + 3a 2 Z> + 3a 2 c + 3b?a + 3Z> 2 c + It is desirable to introduce here some examples of the application of the process of the substitution of a letter for any number or fraction to the properties of numbers, inequalities, &c. 16. Properties of Numbers. Ex. 1. If unity is divided into any two parts, the dif ference of their squares is equal to the difference of the parts themselves. Let x stand for one part ; 1 - x for the other. Now (1 - xf - a 2 = 1 - 2x + x 2 - x 2 = 1 - 2x - (1 - x) - x. i.e., the difference of the squares of the parts is equal to the difference of the parts. Ex. 2. The product of three consecutive even numbers is divisible by 48. Let 2n, In + 2, 2n + 4, be the three numbers . . their pro duct is 8n(n + l)(n + 2). Now, of three consecutive numbers, n, n + l, n + 2, one must be divisible by 2, and one by 3, /. n(n +!)( + 2) is divisible by 6, whence the pro position. Ex. 3. The sum of the squares of three consecutive odd numbers, when increased by 1, is divisible by 12, but never by 24. Let In- 1, In + 1, 2n + 3, be the three odd numbers.
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