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3d Remainder

0. In this quotient, the coefficient a + b of x2, the coefficient a-b of x and the terın a + b are successively obtained by dividing the coefficient a? — 62 of 23 in the dividend, the coefficient a? — 2 ab + b2 of 22 in the first remainder, and the coefficient -a? + 2 ab 62 of x in the second remainder, by the coefficient a -b of x in the divisor.

Ans. (a + b) z2 + (a−b)x — (ab). 2. Divide (6 6 - 10) a4 — (7 62 — 23 6+20) a3 — (363 - 22 62 +31 6 — 5) a2 +(4 63 — 962 +56 —5) a + 62 - 2 b by (3 b—5) a + b2 -26.

Ans. 2 a3 — (3 b — 4) a2 +(46— 1) a + 1. 3. Divide — 28 — (62— 2 c2) a1 + (64 — (4) a? +(66+ 2 64 c + b2c4) by a? - 62 - c.

Ans. -at - (2 62—(2) a? - 64 - 62 cm.

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Terms of a fraction may be multiplied or divided by the same quantity.

CHAPTER II.

FRACTIONS AND PROPORTIONS.

SECTION I.

Reduction of Fractions.

54. 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.

55. Theorem. The value of a fraction, or of a ratio, is not changed by multiplying or dividing both its terms by the same quantity.

Proof. 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. 35, 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.

56. The terms of a fraction can often be simplified

Greatest Common Divisor.

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.

57. Problem. To find the greatest common divisor of several monomials.

Solution. It is equal to the product of the greatest common divisor of the coefficients, by those different literal factors which are common to all the monomials, each literal factor being raised to the lowest power which it has in either of the monomials.

58. EXAMPLES.

a 2

1. Find the greatest common divisor of 75 78 c dll 29 and 50 a3 c? dll 25.

Ans. 25 a3 c dil 25.

121 a 62 c3 d4 25 y6 2. Reduce the fraction

to its lowest 132 a6 65 c4 d3 x2

11 d x3 y5 terms.

Ans.

1:2 a5 63 c'

17 a3 b 3. Reduce the fraction

to its lowest terms. 51 a 65

Ans.

3 64 59. 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

Greatest Common Divisor.

common divisor of A and B be D, and that of B and R be E. We are to prove that

D= E. Now since R is the remainder of the division of A by B, we have

R=A-B.Q; and, consequently, D, which is a divisor of A and B, must divide R; that is, D is a common divisor of B and R, and cannot therefore be greater than their greatest common divisor E. Again, we have

A= R +B.Q and, consequently, E, which is a divisor of B and R, must divide A ; that is, E is a common divisor of A and B, and cannot therefore be greater than their greatest common divisor D.

D and E, then, are two quantities such that neither is greater than the other; and must therefore be equal.

60. Problem. To find the greatest common divisor of any two quantities.

Solution. Divide the greater quantity by the less, and the remainder, which is less than either of the given quantities, is, by the preceding article, divisible by the greatest common divisor.

In the same way, from this remainder and the divisor a still smaller remainder can be found, which is divisible by the greatest common divisor ; and, by continuing this process with each remainder and its corresponding divisor, quantities smaller and smaller are found, which are all divisible by the greatest common divisor, until at length the common divisor itself must be attained.

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