Elementary algebra, with brief notices of its history - Softcover

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1879 Excerpt: ... multiple of two or more algebraical quantities is that quantity which is the least multiple of each of the given quantities; and the least common multiple is the least multiple of each of the given quantities respectively; and is also the least quantity that is exactly divisible by each of the given quantities. Algebraical quantities are said to be prime or composite, in the same manner as numbers in arithmetic. 2. Prop. To explain the process of finding the highest common divisor of two algebraical expressions. The process depends on these two principles. 1. If one quantity be divided by another, and leave a remainder, and if this remainder and the divisor have a common divisor, then this common divisor will also be a common divisor of the dividend and divisor. Let the quantity A when divided by B give the quotient q and remainder r; and suppose p to be the common divisor of B and r, so that--= m and-= n, and therefore B = mp and r = np, or B and r are P P multiples of p. But--= q+L or A = Bq+r. Hence A = mpq+np-pmq--n). A,.--= mq+n, or j» is a common d P 2. That a dividend may be multiplied by any factor which is not See the Editor's Elementary Arithmetic, Section viii., on Measures and Multiples. common to a divisor, and a divisor may be divided by any factor which is not common to a dividend: for in both cases the common divisor of the two expressions is not affected by such operations. The highest common divisor of Aa and B, of A and Bb, and of Aa and Bb, is the same as that of A and B, if a contain no divisor of Bb, nor b of Aa. Let A and B denote two polynomials of which the dimensions of B are not higher than those of A. Let A divided by B give a quotient P, with a remainder Cc. Supposo the factor c in Cc not contained in B, let c be rejected ...

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