a. a. a or R DEF.—A function of x is an algebraical expression involving w; and a rational integral function of u is an expression of the form ax" + boch -1 + &c. + SIC + t, where all the powers of ac are integral and positive. Let f (x)* be a rational integral function of x, and suppose @ to be the quotient, and R the remainder on dividing the function by a Then, evidently l (- a) + R = f (x) identically. And this identity must hold for all values of x, and therefore holds when 3 = In this case we have Q (a a) + R = f (a) Ó + R = f (a) 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 x - 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 2 203 6 x + 7 by a 2 is 15. For, putting a = 2, we have— 2 203 5 x + 6x + 7 2.23 5.22 + 6.2 + 7 15. Ex. 2. The function 2c3 2 2 + 5 x 52 is divisible by a 4. For, putting a = 4, we have 2c3 2 cca + 5 x 52 = 43 - 2.42 + 5:4 52 = 0. COR. 2. Any rational integral function of x is divisible by 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 x - 1. Ex. Each of the following functions is divisible by 2C - l, viz.: 3 c4 + 7 2c3 2a + 12 a 3, (a - b) a + (6-c) x + (c - a), (a + b)*** - 4 abs - (a - b)? * The expression f (x) must not be considered to mean the product of f and x, but as a symbol used for convenience. 21, 5 x? 2 30 Put a = + 1. CoR. 3. Any rational integral function of x is divisible by C + 1, when the sum of the coefficients of the even powers of 3 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 axch + axh -1 + &c. + race + sx + t be a rational integral function of x. 1, then we have, if the function be divisible by 3 + 1 a (-1)" +B(-1)*-' + &c. + F(-1)* +8 (-1) + t = 0. Suppose n to be even, then evidently ( – 1)" : ( - 1) (-1) (-1)...to an even number of factors And so ( - 1)* -1 = ( - 1) ( - 1) (-1)...to an odd number of factors - 1; and so on. Hence we get a 6 + &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 a + 1, viz.:23 + 5 x + 7 2 + 3, 5 206 4 oct + 8 m2 1, a act (a + 1) (a + 2) ** + 2 x + 3 a +4, poco + (q + r) 3cm +(q + r) x + p. Ex. IX. Resolve into elementary factors, 1. ** - 9 a', 16 y* – 25 %, 24 a- 54 6", 8 2 – 27 y. 2. a* - wy*, as - 68, acy* + x*y, 2 a*yoz - 8 xy*x*. 3. a* - 4 6*, ** + y + y, a* – 2 a'b+ 54, a’ + 62 - 6+ 2 ab. 4. a? + 62 - c - d? + 2 ab – 2 cd, a? 72 - C + d + 2 bc + 2 ad, a' - (6 - c). 5. (c + 7)2 – (2 + 2), (2 + 5)2 – ( + 2), (2 a + b)* - (a – c)? 6. (203 - y)2 + 4 (2* + ac y + y) a*y, (2 + y) 5 (ac* + y)2 x*ya + 4 x^y^. 72, 7. 2 - 3 2 - 70, 30 + 11 a + 10, a' - 15 ab + 56 6", och - 420 192. 8. a fac + abocy - 42 b'y', 3 ax – 24 ax – 60 a, 24 ac 5 ac + ac. 9. 6 oca 11x – 35, 8 x + 6 x 135, 1832 212 20 x 42. 10. 3 ay + 10 x*y* + 3 ay, 20 200 + 12 ax + 25 bar + 15 abx, mʻz + (mq + mp) * + pq. Write down the quotient of 11. c* - 16 by – 2, 320 + 96 by a + 2, 220 - 27 by ac - 3. 12. (a + b)" (c + d) by a + b + c + d, a 63 + c - + 2 ac + 2 bd by a + b + c d. Find the remainder after the division of 13. 2c* + pack + qaca + rac + s by s a, act + af by ca a. 14. avd - 5202 + 73 - 9 by x + 3, ** – 3.0 + 7 by . - 2. Show that 15. 52c5 3 203 + 7 22 1 is divisible by a - 1. 16. 2 act 3 x3 + x2 13 is divisible by x + 1. 82 7 2C 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 formis 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 + ... ton terens coefficient of the term to the power in question, and multiply ing the exponents of the letters of the term by the exponent of the power in question. Thus, (a3)2 = a* x a3 = a3 + 3 ?, (am) = am xam xam......to n factors = a mn And so, (4 a?b%c)3 = 48 a2x 378*34*8 = 64 a%bd%!. 33. Development of the third, fourth, and fifth powers of a +b. We know (Art. 26, I.) that (a + b)2 = a2 + 2 ab + 6*. .: (a + b) = (a + 2 ab +62) (a + b) = a + 3 ab + 3 ab + b.; also, (a + b)* (a® + 3 ab+ ab + b) (a + b) a + 4 aRb + 6 a2b2 + 4 abs + 64; and (a + b)" (a* + 4 a3b + 6 a2b3 + 4 abs + b) (a + b) a' + 5 a+b + 10 a°62 + 10 aʼ53 + 5 ab* + 65. The following law of the formation of the terms is evident: + + + Law of Formation of Terms. 1. The first term contains a raised to the given power, and the power of a decreases by unity in each successive term, while the power of b (which first appears in the second term) increases by unity in each successive term, till it reaches the power of the given quantity. 2. The first coefficient is unity, and the coefficient of any term is found by multiplying the previous coefficient by the exponent of a in the previous term, and dividing the product by the number of terms bitherto developed. Ex. 1. (2 x + 3y)' = (2x)* + 3 (2x)" (3 y) + 3 (2 x) (3 y). + (3 y) = 8.200 + 36 xy + 54 xy + 27 y. Ex. 2. (a + b + c) (a + b + c) = (a + b)3 + 3 (a + b)2 C + 3 (a + b) c + os (as + 3 a+b + 3 ab2 + 65) + 3 (a? + 2 ab + 6?) c + 3 (a + b)(a + c3 as + 73 CS + 3 a+b + 3 a'c + 3 ab + 36%c + 3 ac + 3 bc + 6 abc. (In the following examples the above law may be assumed as generally true.) Ex. X. 1. Find the values of (a*b), (- 3 ab), (2 a+bc), (- a*yz?)? Expand- 4. (2c2 + + 1), (3 a b + 4c - d), (a + 2b - c)? (3a + 3b + 3c)? 5. (1 + 2 - **)", (ax + by + cz)",{(a+b)x – (c + d) y}: 6. (1 + x), (1 + 2 + 2*)*, (a + bxc + c«c) 7. (ax – by), (3 x + y): (3x – y), (2c + xy + y)* (– y)”. 8. (x* + ac* y* + y^)? (+ y)2 (2 {(a + b)? – 4 ab}'{la 4 ab}'{(a - b)* + 4 ab }? Simplify9. (a + b + c): – 3 (a + b + c)* c + 3 (a + b + c)co - . 10. (a - b)+ 3 (a b)? (6 c) + 3 (a - b)(b - c) + (6 - c)? 11. (1 + x + 32 + 3 200)* + (1 – 2 + 3 ** – 3200)?? 12. {(z + 3) - (x + y)} - 27 2ư ( + y). y)', Evolution. 34. Evolution is the operation by which we obtain the roots of quantities. Since the square or second power of a® is (a)? or a', we call (Art. 17) as the square root or the second root of a®. And so, since the cube or third power of a’ is (a?)or a®, we call a' the cube root or the third root of ar. So, generally, since the nth power of a". (am)" = ann, we call am the nth root of amn. Thus, we have vão a’, vam = a". Hence, in the case of quantities consisting of a single letter with a given exponent, when the given exponent contains as a factor the number indicating the root, we must divide the given exponent by this number, the quotient being the exponent of the root. (We shall see farther on that this rule holds when the given exponent is not so divisible, the root in this case being called a surd). mn |