Hence we derive the following RULE. Divide the coefficient of the dividend by the coefficient of the divisor, and strike out the letters of the divisor from the divi dend. Observe that 4 a3 is the same as 4 a a a and a2 is the same as a a; 4 a a a divided by a a gives 4 a for the quotient. It was observed in multiplication, that when the same letter enters into both multiplier and multiplicand, the multiplication is performed by adding the exponents, thus a3 multiplied by a is a3+2 = a3. In similar cases, division is performed by subtracting the exponent of the divisor from that of the dividend. a divided by a2 is a -2 == α. The division of some compound quantities is as easy as that of simple quantities. If a+b+c be multiplied by d the product is d(a+b+c) or ad+bd+cd. Therefore if a d+bd+cd be divided by d, the quotient is a+b+c. If ad+bd+c d be divided by a+b+c, the quotient is d. When a compound quantity is to be divided, let the quantity, if possible, be so arranged that the divisor may appear as one of the factors, and then that factor being struck out, the other factor will be the quotient. 19. Divide 12 a2b-9 ac by 3 a. 12 ab-9ac3a (4ab-3c) Ans. 4ab-3c. Observe that a is a factor of both terms, and also 3. Hence the quantity 12 ab-9 a c, can be resolved into factors; thus 3(4 ab-3 a c), or a (12 ab-9 c), or 3 a (4 ab-3c). In the last form the divisor 3 a appears as one factor, and the other factor 4 ab-3 c is the quotient. Note. Any simple quantity, which is a factor of all the terms of any compound quantity, is a factor of the whole quantity; and that factor being taken out of all the terms, the terms as they then stand, taken together, will form the other factor. 20. Divide 8 a b3 — 16 a3 bc by 2ab-4a3 c. 8 a2 b3 — 16 a3 b2 c4 a b2 (2 a b — 4 a3 c.) 21. Divide Ans. 4 abs. 3 a b c - 15 a b2 d + 9 a3 b d2 by 3 a b. 22. Divide 15 abc-30 ac2 + 25 a3 c d by 11 5 a2 c. 23. Divide 36 a13 bc-28 a11 b c2 + 40 ao be c3 24. Divide 42 a84 a10 b c by 1-2 a3 b2 c. Algebraic Fractions. XV. When the dividend does not contain the same letters as the divisor, or but part of those of the divisor, the division cannot be performed in this way. It can then only be expressed. The usual way of expressing division, as has already been explained, is by writing the divisor under the dividend in the form of a fraction. Thus a divided by b is expressed α This gives rise to fractions in the same manner as in arithmetic. It was shown in arithmetic, that a fraction properly expresses a quotient. Algebraic fractions are subject to precisely the same rules as fractions in arithmetic. Many of the operations are more easily performed on algebraic fractions. In these, as in arithmetic, it must be kept in mind, that the denominator shows into how many parts a unit is divided; and the numerator shows how many of those parts are used; or the denominator shows into how many parts the numerator is divided. I shall here briefly recapitulate the rules for the operations on fractions, referring the learner to the Arithmetic for a more full developement of their principles. of 7 is; for } of 7 is a is 24; for 4 of a is, and α b = ac 1, and is 3 times as much. of is 2 times as much. The part of c is; for of c is, and is a times as much. b Hence, to multiply a fraction by a whole number, or a whole number by a fraction, multiply the numerator of the fraction and the whole number together, and divide by the denominator. Arith. Articles XV. & XVI. This cannot be done like the others, but it may be done by multiplying the denominator as in Arith. Art. XVII. For the fraction denotes, that one is divided into as many equal parts as there are units in b, and that as many of these parts are used as there are units in a ; or that a is divided into as many equal parts as there are units in b; hence if it be divided into twice as many parts, the parts will be only one half as large, and the fraction will have only one half the value. |