CHAP. V. ON THE REDUCTION OF ALGEBRAICAL EXPRESSIONS TO EQUIVALENT AND MORE SIMPLE FORMS. of alge terms have 167. In the division of algebraical quantities, where Reduction the quotient would not be complete and finite, it is com- braic fracmonly most convenient to leave the dividend and divisor tions, whose under a fractional form: but in many cases such a form a common would not be the most simple which the expression is divisor. capable of receiving, without altering its value and signification: inasmuch as the numerator and denominator may be divided by any factor, whether simple or compound, which is common to them both. factor, 168. If this factor or common divisor be a simple A common Algebraical term, it is discoverable by inspection, and the which is a reduction required may be effected at once: thus, a is a single term. divisor of every term of the numerator and denominator of and other instances will present themselves amongst the examples which follow. Again, if all the terms of the numerator and denominator A numerihave numerical coefficients, a common measure may exist cal factor. A com pound factor. The process for finding amongst them, which may be detected either by inspection or by means of the rule (Art. 152), the proof of which has been given in the preceding Chapter. 169. But the reduction of an algebraical fraction which is effected by these means, is frequently not the only one of which it admits: for in many cases the numerator and denominator may have a common factor consisting of more than one term, which is not discoverable by inspection for as the factors of a numerical product are lost sight of in the result, so likewise an algebraical factor consisting of more than one term, after its incorporation with other quantities by multiplication, may leave no discoverable traces of its separate existence: it becomes an enquiry, therefore, the importance of which is not limited to the reduction of algebraic fractions, to find the factor of the highest dimensions which is common to two algebraical expressions: we say factors of the highest dimensions and not the greatest, for the terms greater and less can have no application when we are reasoning with general symbols, to which no specific values are given. 170. The process for finding such compound factors compound is analogous to that for finding the greatest common measure of two numbers. factors. Let A and B represent two compound algebraical expressions: arrange them as much as possible according to the powers of some one letter, and make that quantity the divisor whose dimensions are not the highest let the remainder after division be Ce, where C is a compound quantity, and where c is a quantity whether simple or compound, which is obviously not a divisor both of A and B: make C alone the new divisor and the last divisor the new dividend, and let the remainder, if any, after division, be Dd, where d is not a factor of A and B: make D a new divisor and Ca new dividend: if there is no remainder after this division, then D is the required compound factor of A and B: if there is a remainder, we must proceed as before until a divisor is found which leaves no remainder: if no such divisor can be found, A and B have no common factor, or this process fails to discover it. The following scheme may help to make the process more intelligible : B) A (P PB Cc C) B (Q QB Dd D) C (R ᎡᎠ In order to prove the truth of this process, we must premise the following Lemma. 171. LEMMA. If D is the highest common divisor of Lemma. A and B, it is likewise the highest common divisor of Aa and Bb, if a and b have no common factors. For the factors which are common to A and B, are the same which are common to Aa and Bb, no new common factor having been introduced by multiplying a into A and b into B. It follows from this Lemma, that we may divide or multiply either A or B, by a factor which is not common to both, or which has no divisor common to both, without affecting the dimensions or form of their highest common factor. 172. That D, the last divisor found by this process, Proof of is a factor of A and B, may be proved as follows. the rule. A-PB+Cc. B=QC+Dd. C=RD. D is a divisor of C, and therefore of QC and Dd, and therefore of QC+ Dd or B: and since it is a divisor of B and C, it is a divisor of PB and Cc, and therefore of PB+Cc or A. Again, every compound divisor of A and B is a divisor of D. For every compound factor of A and B is a factor of A and PB, and therefore of A-PB or Cc, and therefore of C alone, since c has no factor common to A and B : it consequently divides B and QC, and therefore B-QC or Dd, and therefore D alone, since d has no factor common to A and B. It follows, therefore, that D is the highest compound divisor of A and B. The highest common divisor, therefore, of — a3 and x2-a2, is x-a, and the fraction reduced is x2 + a x + a2 x + a In this example we have stopped the first division after one operation: if we had proceeded further, the successive terms in the quotient would have been a2 a3 &c. and the successive remainders would have been quantity x-a for the new divisor: in order to avoid such unnecessary divisions, it is convenient to adopt this general rule: continue the division as far as possible in each case Rule for without introducing a term into the quotient under a unnecessary fractional form, and no further. avoiding divisions. (2) Reduce x2+(ab)x-ab to its lowest terms. When the first terms of each expression, arranged according to the powers of some one letter, are identical, write one of them beneath the other and subtract. The common factor is r+a, and the fraction reduced is x-b x+b |