CHAPTER XIV. Reduction of single fractions. ON THE REDUCTION OF ALGEBRAICAL EXPRESSIONS TO THEIR 621. WHEN an algebraical expression presents itself under a fractional form, whose numerator and denominator admit of a common divisor, their dimensions may be lowered and the form of the fraction generally simplified without altering its signification or value, by the process which is given in the last Chapter. 622. When two or more algebraical expressions, one or more of which are under a fractional form, are connected by tions, con- the signs + and −, they may be reduced to their lowest common nected by denominator, and subsequently added or subtracted or reduced to their most simple equivalent form, by the same rule, mutatis mutandis, which is given for the addition and subtraction of numerical fractions in Art. 124: it is as follows. Reduction of two or more frac the signs + and - to their most simple equivalent form. Rule. Examples. RULE. "Find the LOWEST common multiple (Art. 619) of the denominators of all the fractions, for the new common deno minator. Find the successive quotients which arise from dividing this LOWEST common multiple by the several denominators of the fractions and multiply them successively into the numerators of the several fractions, thus forming the successive numerators of the equivalent fractions with a common denominator; connect the new numerators together with their proper signs, and beneath the result write the common denominator, reducing the fraction, which they form, to its lowest_terms.”* 623. The following are examples. (1) To reduce the expression equivalent form. a + b a-b + a- b a + b to its most simple The lowest common multiple of the denominators is their product a2-b2: x+6 (2) To reduce the expression x2+4x-21 most simple equivalent form. The lowest common multiple of the denominators (D) is = x2 + 1 4(x+1)(x+3) (x + 1)3 (x+3)' which admits of no further reduction. In the preceding reductions we have adhered strictly to the general rule, though the same result can frequently be obtained by shorter and more expeditious processes, which it is not necessary to notice: they can only be safely employed by a student who has already become familiar with algebraical operations, and whose memory is stored with an habitual knowledge of a great number of their more simple and elementary results*. division of 624. The rule for the multiplication and division of fractions Multipliis given in Art. 143: and it will be seen, by a reference to the cation and Articles which precede it, that it is derived by a species of an- fractions. ticipation, from the principles which are made the foundation of Symbolical Algebra: those principles, as applicable to the cases under consideration, may be restated as follows. Supposing to be the multiplier, we may assume that amongst the successive values of c there is one which is equal to md, с where m is a whole number, making equal to a under such circumstances, therefore, the product of and b am becomes the product of and m, which is (Art. 130): b such would be the result if the general form of the product of b с ac and was and no other form of this product will satisfy bd' the required condition: and inasmuch as it is assumed that the form of this product, whatever it may be, is independent of the specific values of the symbols involved (Art. Supposing the divisor, as before, to become or m, where m is a whole number, it will follow that the quotient of by will become, under such circumstances, identical with (9) (10) (11) d (b − u) ( c − c ) ( x + a ) † ( a − b ) ( c − b ) ( x + b) * ( a − c ) ( b − c ) ( 1 + c) Quotient of one fraction divided by another. Examples. a a the quotient of b by m, which is (Art. 131): such would bm a C be the result, if the form of the quotient of by was : ad bc' and no other form of this quotient will satisfy the required condition and inasmuch as it is assumed that the form of this quotient, whatever it may be, is independent of the spead cific values of the symbols involved, it follows that which a bc > is the form of the quotient of divided by in one case, must be its form likewise in all others. d 627. The following are examples of the multiplication and division of fractions. x + 4 x + 3 =3 = (x2 − 9) (x − 1) __ x23 − x2 − 9x + 9 when reduced to its lowest terms. = a3 — aob + ab2 – b3 a3 + a2b- ab2 - b3 when reduced to its lowest terms. a2 + b2 a - b _ (a2 + b2) (a + b) _ a3 + a2b÷ab2 + b3 a2 + b2 a2 - b2 ̄ ̄ a+b ̄ ̄ (a2 — b2) (a−b) ̄ ̄ a' — a`b — ab2 + b3 ̄ ̄ a2 — 2ab+b2' when reduced to its lowest terms. 628. The quotient of the division of a by tiplication the product of a and a 1 is ab, and is (Art. 144.): it follows, therefore, |