w? – – 30 = (x – 6) (x + 5); for the product of - 6 and 5 is - 30, and their sum is - 1. And so, ac* - 18 x + 32 = (x - 2) (x - 16), And a + 3 ab - 108 2 = a + (12 6 - 9 b) a + (126) (- 96) = (a + 126) (a – 96). (IV.) The form axa + bxc + C. This is the general form of a trinomial. The following remarks, though equally applying to each of the three preceding forms, are especially intended to be practically applied to trinonnials not included by them. The above form will include such expressions as the following:—20 x2 + 11 2 – 42, 6 ac – 37 X + 55. It is evident that the product of the first terms of the factors will be the first term of the given trinomial, and that the product of the last terms of the factors will be the third term of the given trinomial. And, further, when the third term is negative, the last term of one factor must have the sign +, and the last term of the other the sign -; but, when the third term is positive, the last terms of the factors must have the same sign as the middle term. Thus, 12 x2 – 31 -' 30 = (4 x + 3) (3x – 10). Here the factors of 12 ca are either 3 x and 4 x, 6 x and 2 x, 12 x and x, and the factors of 30 either 5 and 6, 3 and 10, 2 and 15, 1 and 30; and we must give a + sign to one of each of these latter pairs, and a – sign to the other. It is easily found on trial that, in order to obtain - 31 x as the middle termi, the factors of the trinomial must be 4 x + 3 and 3 x – 10. So we have 10 a- 41 ab + 21 72 = (5a – 36) (2 a -76), and aca2 + (ad + bc) xy + bdij = (ax + by) (ca + dy). (V.) The forms x" + yr and 2" – ym. We shall show in the next article that a rational in. tegral algebraical expression, involving x, contains a – a as a factor when it vanishes on substituting a for X. Hence, ach + y" and 2" - y" must each vanish on putting y for x, if they contain x – y as a factor, n being an integer. The former becomes y" + y" or 2 y", and the latter yn y" or 0. We therefore conclude that Para + yn does not contain x - y as a factor, and that, 20n - yn does contain x - y as a factor, whether n be even or odd. Again, on the same principle, they must each vanish if they contain x + y as a factor, on putting – y for X. Theformer becomes( - y)” + y", which vanishes when n isodd, and thelatter becomes (-y)” – y”, which vanishes when n is even. Hence we conclude that cm + y" contains x + y as a factor when n is odd, and och – yn contains c + y as a factor when n is even. Now, the quotient of either of these quantities by x + y or 3 – y can in any particular case be found by long division. We thus find that—. 200 + y = (x + y) (24 – xy + 2x^y– xy + y^), 203 - Y3 = (oc – y) (2c2 + xy + y^). The law of formation of the co-factor in each case is easy to see; and if we may assume this apparent law as generally true, we may conclude that, when an algebraical quantity is of the form 2cm + y or 2c* - y", and it contains x + y or x - y as a factor, the law of formation of the co-factor is as follows: Law of Formation of Co-Factor. 1. The terms are homogeneous, and of dimensions one degree lower than the given expression, the power of w in the first term being n - l, and diminishing each successive term by unity; and the power of y increasing each successive term by unity, and first appearing in the second term. 2. The coefficient of every term is unity. 3. The signs are alternately + and – , when u + y is the corresponding elementary factor; and are all +, when X – Y is the corresponding elementary factor. Ex. 1. ax + 32 = ab + 25 = (a + 2) (a* + a3 • 2 + a2 • 22 + a. 23 + 24) = (a + 2) (a* + 2 a3 + 4 a2 + 8 a + 16). Ex. 2. am - 78 = (az)2 – (63)2 = (a3 + 63) (@3 - 63) = (a + b) (a- ab + 62) : (a - b) (a2 + ab + b2 = (a + b) (a - b) las - ab + b2) (a2 + ab + 6%). 30. The remainder of the division of a rational integral function of x by x - a may be found by putting a for x in the given function. DEF.-A function of w is an algebraical expression involving w; and a rational integral function of x is an expression of the form ax+ bxh 1 + &c. + 8C + t, where all the powers of x are integral and positive. Let f (x)* be a rational integral function of x, and suppose Q to be the quotient, and R the remainder on dividing the function by x - a. Then, evidently Q (20°– a) + R = f (x) identically. And this identity must hold for all values of x, and therefore holds when x = a. In this case we have Q (a - a) + R = f (a) 0 + R = f (a) or R = f(a). Now f (a) is the result of putting a for x in the given function, and is, as we have just shown, the remainder on dividing the given function by a - a. COR. 1. When there is no remainder, we must, of course, have f (a) = 0. Hence, a given rational integral function of x vanishes when a is put for x, if it be divisible by X - a. Ex. 1. The remainder, after the division of 2 2 - 5 m + 6 x + 7 by 2 – 2 is 15. For, putting a = 2, we have 2 203 – 5 mca + 6x + 7 = 2• 23 – 5 • 22 + 6.2 + 7 = 15. Ex. 2. The function 23 – 2 202 + 5 3 – 52 is divisible by x – 4. For, putting x = 4, we have 2003 – 2 x + 5 x – 52 = 48 – 2 · 42 + 5 • 4 – 52 = 0. COR. 2. Any rational integral function of x is divisible by 2 – 1, when the sum of the coefficients of the terms is zero. For, putting x = 1 in the given function, it is evident that it is reduced to the sum of its coefficients, which sum must be zero if the function be divisible by a – 1. Ex. Each of the following functions is divisible by 2 – 1, viz.: 3 2* + 7 203 – ac2 + 12 x – 21, 5 x? — 2 3 – 3, (a - b)2c2 + (6—c) x + (c – a), (a + b)* x* — 4 abx-(a - b)?. * The expression f (2) must not be considered to mean the product of f and X, but as a symbol used for convenience, COR. 3. Any rational integral function of x is divisible by XC + 1, when the sum of the coefficients of the even powers of a is equal to the sum of the coefficients of the odd powers. (The term independent of x is always to be considered as the coefficient of an even power). Let ax" + ax" – 1 + &c. + rx + 8x + t be a rational integral function of . Put a = - 1, then we have, if the function be divisible by x + 1 a (-1)" + b (-1)" –1 + &c. + 5 (-1): + 8 (-1) + t = 0. Suppose n to be even, then evidently ( – 1)" = (-1) ( - 1) (-1)... to an even number of factors = +1. And so ( - 1)" – 1 = ( - 1) ( - 1) ( – 1)...to an odd number of factors = -1; and so on. Hence we get a – b + &c. + p – 8 + t = 0, and this must evidently require the condition that the sum of the positive quantities is equal to the sum of the negative, and, therefore, that the sum of the coefficients of the even powers of x is equal to the sum of the coefficients of the odd powers. And a similar result will follow if we suppose n to be odd. Ex. Each of the following functions is divisible by oc + 1, viz. : 2x3 + 5 22 + 7 x + 3, 5 206 — 4 x4 + 8 ac – 2 x – 1, aRxc4 – (a + 1) (a + 2) 2* + 2 x + 3 a + 4, pac + (q + r) 22 +(2 + r) ac + p. Ex. IX. Resolve into elementary factors, 1. 2 – 9 aʻ, 16 y4 – 25 A, 24 a’ – 54 62, 8 203 – 27 y. 2. 2* — xyo, a8 – 65, xy* + x^y, 2 x*yoz – 8 xy+z3. 3. a* – 4 64, 2c4 + x*yo + y^, at – 2 a?b% + 64, a2 + 22 – co + 2 ab. 4. a2 + 62 — * -+ 2 ab – 2 cd, a’ - 62 – + d + 2 bc + 2 ad, a? – (6 - c). 5. ( + 7) – (2 + 2)?, (oc + 5)2 – (c + 2), (2 a + b)2 - (a – c)? 6. (23 - 2)2 + 4 (2* + x*yo + y^) xy, (x2 + y) 5 (2c2 + y)2 ya + 4 x*y*. M 7. do - 3 g - 70, ° + 11 z + 10, do - 15 ab + 56 6°, có - 4X – 192. 8. a*x + abxy - 426*y*, 3 axe – 24 ax – 60 a, 24 ac – 5 ac“ + acé 9. 6 ** - 11x – 35, 8 xo + 6 x – 135, 18x* – 21x - 72, 20 22 – 11 2 – 42. 10. 3x®y + 10 xʻya + 3 xy?, 20 203 + 12 ax? + 25 bxo + 15 abx, m-az + (mg + mp) 2 + pq. Write down the quotient of 11. ** - 16 by x – 2, 3 x3 + 96 by x + 2, 20% - 27 by 22 - 3. 12. (a + b)2 – (c + d) by a + b + c + d, a – 12 + c - d + 2 ac + 2 bd by a + b + c - d. Find the remainder after the division of 13. ** + pacs + quca + rac + 8 by - a, 20* + at by 2cm - a?. 14. 23 – 5x2 + 72 – 9 by x + 3, ** – 3x + 7 by x – 2. Show that 15. 5.205 - 3 203 + 7 22 – 8 x – 1 is divisible by a – 1. 16. 2 - 3 23 + x2 – 7 x – 13 is divisible by x + 1. CHAPTER III. INVOLUTION AND EVOLUTION. Involution. 31. Involution is the operation by which we obtain the powers of quantities. This can of course be done by multiplication, but the results obtained by the actual multiplication of simple forms enable us to develop without multiplication more complex forms. As the subject requires the aid of the Binomial Theorem, we shall here show how to develop a few only of the more simple expressions. -32. The power of a single term is obtained by raising the |