Difference of two Powers divisible by Difference of their Roots. 37. 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 - bn by a-b, as in art. 34, proceeding only to the first remainder, as follows. _b, 1st Remainder an-1b — bn = b (an — 1 — bn —1). Now, if the factor an—1— bn-1 of this remainder is divisible by a―b, the remainder itself is divisible by a and therefore an-b" is also divisible by a proposition is true for any power as the holds for the nth, or the next greater. -b; that is, if the (n - 1)st, it also But from examples, 10, 11, 12, 13 of art. 34, 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. 38. 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. It 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. 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 Division of Polynomials. to arrange the terms in the vertical columns according to the powers of some letter common to them. EXAMPLES. 1. Divide a2 x3-b2x3-4 a b x2-2 a2x+2abx+ a2 -b2 by ax· - b x Ans. (a+b) x2 + (a — b) x − (a — b). 2. Divide (66-10) a1 — (7 b2 — 23 b+20) a3 — (363 -22 b2+31 b—5) a2 + (4 b3 —9b2+5b—5) a + b2 -26 by (36- 5) a + b2 — 2 b. Ans. 2a3-(3b-4) a2 + (4 b—1) a +1. Division of Polynomials. 3. Divide a6 (b2 — 2 c2) a2 + (b1 — c1) a2 + (b 6 + 2b4c2b2c4) by a2-b2-c2. Ans. a-(2 b2 — c2) a2 —b4—b2 c2. Terms of a fraction may be multiplied or divided by the same quantity. CHAPTER II. Fractions and Proportions. SECTION 1. Reduction of Fractions. 39. When a quotient is expressed by placing the dividend over the divisor with a line between them, it is called a fraction; its dividend is called the numerator of the fraction, and its divisor the denominator of the fraction; and the numerator and denominator of a fraction are called the terms of the fraction. When a quotient is expressed by the sign (:) it is called a ratio; its dividend is called the antecedent of the ratio, and its divisor the consequent of the ratio; and the antecedent and consequent of a ratio are called the terms of the ratio. 40. Theorem. The value of a fraction, or of a ratio, is not changed by multiplying or dividing both its terms by the same quantity. Demonstration. For dividing both these terms by a quantity is the same as striking out a factor common to the two terms of a quotient, which, as is evident from art. 30, does not affect the value of the quotient. Also multiplying both terms by a quantity is only the reverse of the preceding process, and cannot therefore change the value of the fraction or ratio. Greatest Common Divisor. 41. The terms of a fraction can often be simplified by dividing them by a common factor or divisor. But when they have no common divisor, the fraction is said to be in its lowest terms. A fraction is, consequently, reduced to its lowest terms, by dividing its terms by their greatest common factor or divisor. 42. The greatest common divisor of two monomials is equal to the product of the greatest common divisor of their coefficients by that of their literal factors, which last is readily found by inspection. EXAMPLES. 1. Find the greatest common divisor of 75 a b c d 11 x9 and 50 a3 c2 dll x5. Ans. 25 a3 cd11 x5. 43. Lemma. The greatest common divisor of two quantities is the same with the greatest common divisor of the least of them, and of their remainder after division. Demonstration. Let the greatest of the two quantities be A, and the least B; let the entire part of their quotient after division be Q, and the remainder R; and let the greatest |
