526 Ex. 5. Divide 12x> J -10* 3 y- A L G E B B A [INVOLUTION AND by orner s We will employ this example to indicate Homer s nthetic method of synthetic division. vision. j^ the dividend be represented by -:- E, the divisor by ax" + bx + c, and the quotient by ax 2 + fix + y + &c. Then, multiplying the quotient by the divisor, we produce the dividend, which, exhibited by the method of detached coefficients, stands thus - aa + a/3 + ay + &c. + ca A+ B + C +&c. The last line being the sum column by column of the three preceding lines. Now, as the upper of these three lines contains term by term the quantities required, we convert this addition into subtraction ; thus, -I A +B +0 +D +E bab/3by &C. - co. - c/3 - &c. a aa + aft + ay + aS + <ic. The first vertical column gives a ; the second (3, and so on. In the example before us we write, 12-10-3 +30-25 + 4a + 4/3 + 4y + &C. 5a 5/3 &c. + 4 -5 whence 3a = 12 gives a = 4; 3/3= 10 + 4a gives /3 = 2; 3y = - 3 + 4/3 - 5a gives y = - 5. Therefore the quotient required is 4x 2 + 2x - 5. SECT. II. INVOLUTION AND EVOLUTION. 21. In treating of multiplication, we have observed, that when a quantity is multiplied by itself any number of times, the product is called a power of that quantity, while the quantity itself, from which the powers are formed, is called the root. Thus, a, a 2 , and a 3 are the first, second, and third powers of the root a : and in like manner -, . a a- and -5 denote the same powers of the root - . a 3 a But before considering more particularly what relates to powers and roots, it will be proper to observe, that the quantities -, -~ , &c., admit of being expressed under a (t CL d different form ; for, just as the quantities a, a 2 , a 3 , &c., are expressed as positive powers of the root a, so the 111 quantities -, , , &c., may be respectively expressed o/ a* Q/ thus a" 1 , a~ 2 , a" 3 , &c., and considered as negative powers of the root a. This method of expressing the fractions -, , r, as a a- a 3 powers of the root a, but with negative indices, is a conse quence of the rule which has been given for the division of powers ; for we consider - as the quotient arising from the division of any power of a by the next higher power - } for example, from the division of the 2d by the 3d, and so 1 a 2 we have - = ; but since powers of the same quantity are divided by subtracting the exponent of the divisor from that of the dividend (Art. 19), it follows that -^ = a 2 - 3 = a~ l ; therefore the fraction - may also be expressed a 1 a? thus, a" 1 . By considering as equal to r , it will appeal a- a 4 in the same manner that 5 = r = a~ 2 ; and proceeding in a 2 a 4 this way, we gGt = = a~ z , = = a~ 4 , &c., and so on, Cl/ Cl Cl Ct as far as we please. It also appears that unity or 1 may be represented by a, where the exponent is a cypher, for The rules which have been given for the multiplication Defmiti< and division of powers with positive integral exponents ot indc*. will apply in every case, whether the exponents be positive or negative, integral or fractional, provided we assume as the definition of the index in such cases, the law of com bination a m x a" = " +". Involution. 22. Involution is the method of finding any power of any assigned quantity, whether it be simple or compound : hence its rules are easily derived from the operation of multiplication. Case 1. When the quantity is simple. Rule. Multiply the exponents of the letters by the index of the power required, and raise the coefficient to the same power. Note. If the sign of the quantity be + , all its powers will be positive ; but if it be - , then all its powers whose exponents are even numbers are positive, and all its powers whose exponents are odd numbers are nega tive. Ex. 1. Required the cube, or third power, of 2crcc, (2a 2 ^) 3 = 2 x 2 x 2a 2x3 . lx3 = 8a 6 ^ 3 , the answer. Ex. 2. Required the fifth power of - 3a 2 x 3 . ( 3a 2 ^; 3 ) 5 = 243a 10 # 15 , the answer. Ex, 3. Required the fourth power of T, the answer. 4- 1 & 2 y / Q SlWy Case 2. When the quantity is compound. Rule. The powers must be found by a continual multipli cation of the quantity by itself. Ex. 4. Required the first four powers of the binomial quantity a + x. a + x the root, or first power. a? + ax a 2 + 2ax + x 2 the square, or second power. a + x + 2ax 2 a 3 + 3a?x + 3ax 2 + x 3 the cube, or third power. a 4 + 3a 3 x + 3a 2 x 2 + ax 3 + a?x + 3a 2 x 2 + 3ax 3 a* + 4a 3 j; + Ga 2 x~ + 4ax 3 + x* the fourth power. If it be required to find the same powers of a - x, it
will be found, writing x for x, that
