Division of Polynomials. 3 x4m y2n + 3 x2m yan - yon by x3m 7. Divide 26m 3 x2m yn + 3 xm y2n — узп. 3η Ans. x3m +3 x2m yn + 3 xm y2n + y3n. 1 + a3 n3 by -1+an. Ans. 1+an + a2 n2. 9. Divide 2 a2 — 13 a3 b + 31 a2 62 - 38 a b3 + 24 64 Ans. a + a3 b + a2b+ab3 + 64. 44. Corollary. The quotient can be obtained with equal facility by using the terms which contain the lowest powers of a letter instead of those which contain the highest powers. In this case, it is more convenient to place the term containing the lowest power first, and that containing the next lowest next, and so on. This order of terms is called an arrangement according to the ascending powers of the letter; whereas that of the preceding article is called an arrangement according to the descending powers of the letter. 45. Corollary. Negative powers are considered to be lower than positive powers, or than the power zero, and the larger the absolute value of the exponent the lower the power. Thus a5 x-6 - a4 x-3 + a3 + a−1 x + a-2 x2, is arranged according to the ascending powers of z, and according to the descending powers of a. Division of Polynomials. 46. EXAMPLES. 1. Divide a + a2 - a-2-a-4 by a2-a-2. Ans. a2+1+a-2. 2. Divide 4 a4 b6+12 a3b-5 + 9a2b-4b2+2a-2 - a-4b2 by 2a2b-3+3ab-2 — 6-1 + a-2b. Ans. 2 a2b-3 + 3ab-2 + 6-1 - a-2b. 47. In the course of algebraic investigations, it is often convenient to separate a quantity into its factors. This is done, when one of the factors is known, by dividing by the known factor, and the quotient is the other factor. And when a letter occurs as a factor of all the terms of a quantity, it is a factor of the quantity, and may be taken out as a factor, with an exponent equal to the lowest exponent which it has in any term, and indeed by means of negative exponents any monomial may be taken out as a factor of a quantity. 48. EXAMPLES. 1. Take out 3 a2b as a factor of 15 a5 b2 + 6 a3 b + 9 a2 b2 + 3 a2 b. Ans. 3 a2b (5 a2b+2a+3b+1). 2. Take out am as a factor of 3 am + 1+2 am. Ans. am (3 a + 2). 3. Take out 2a365 c as a factor of 6 a 67 c2 + 6 a b c -2ab+2— а2 с. Ans. 2 a3 65 c (3 a3 b2c + 3a-2b3 - a-2b-4c-1+ a-36-5-1-2-12-16-5. 4. Take out b as a factor of an-1b-br. Ans. b (an-1-bn-1). Difference of two Powers divisible by Difference of their Roots. 49. Theorem. The difference of two integral positive powers of the same degree is divisible by the difference of their roots. Thus, an b" is divisible by a-b. Demonstration. Divide an br by a-b, as in art. 42, proceeding only to the first remainder, as follows. 1st Remainder = a2-16-b2 = b(an-1-bn-1). Now, if the factor an-1-br-1 of this remainder is divisible by a - b, the remainder itself is divisible by a - b, and therefore an - b" is also divisible by a - b; that is, if the proposition is true for any power, as the (n - 1)st, it also holds for the nth, or the next greater. But from examples, 10, 11, 12, 13 of art. 43, the proposition holds for the 2d, 3d, 4th, and 5th; and therefore it must be true for the 6th, 7th, 8th, &c. powers; that is, for any positive integral power. 50. Corollary. The division of a" - bn by a - b may be continued for the purpose of showing the form of the quotient, that is, Division of Polynomials. = an-1+an-2b+an-3b2+ &c.... + abn2+0-1, so that each term of the quotient is obtained from the preceding term by diminishing the exponent of a by unity and increasing that of b by unity; and the number of terms is equal to the exponent n. 51. Corollary. If b is put equal to a in the preceding quotient, each of its terms becomes equal to an-1, which gives the peculiar result 52. There are sometimes two or more terms in the divisor, or in the dividend, or in both, which contain the same highest power of the letter according to which the terms are arranged. In this case, these terms are to be united in one by taking out their common factor; and the compound terms thus formed are to be used as simple ones. It is more convenient to arrange the terms which contain the same power of the letter in a column under each other, the vertical bar being used as in art. 17; and to arrange the terms in the vertical columns according to the powers of some letter common to them. 53. EXAMPLES. b2 x3 - 4 a b x2 - 2 a2x + 2ab x + 1. Divide a2 x3 a2-b2 by ax - bx - a - b. In this quotient, the coefficient a + b of x2, the coefficient a-b of x and the term - a + b are successively obtained by dividing the coefficient a2 - 62 of 23 in the dividend, the coefficient a2-2ab + b2 of 22 in the first remainder, and the coefficient - a2 + 2ab - 62 of x in the second remainder, by the coefficient a - b of x in the divisor. Ans. (a+b) x2 + (a - b) x - (a-b). 2. Divide (66 - 10) a4 - (762 - 23 b +20) а3 — (363 2262 +31 6 - 5) а2 + (4 63 - 962 + 56 - 5) a + b2 b -26 by (36-5) a + b2 - 2 b. 3. Divide a2 Ans. 2 a3- (36-4) a2 + (46 — 1) a + 1. a® - (b2 - 2 c2) a2 + (b - c4) a2+(66 + |