41. A product being given, we may sometimes by mere inspection decompose it into its factors, an operation which is frequently useful. Let there be the product a ba2 b2, in the formation of this product each term, it is evident, has been multiplied by a2 and also by b, its factors therefore are a2, b and a b, and it may be put under the form a2 b (a - b) In like manner the product a c + ad÷b c + b d may be put under the form a (c+d) +b (c + d), or which is the same thing (a+b) (c + d). EXAMPLES. 1. To find the factors of the product a2 b2+2 a3 b3 —— a3 b2 c 2. To find the factors of the product 3. To find the factors of the product a3 c + a b c + a2 d2 +b d2 4. To find the factors of the product a4bc-2 a3 b3 c + a d2 a3 b2 d DIVISION OF ALGEBRAIC QUANTITIES: 42. 1. The object of division in algebra is the same as that of division in arithmetic, viz. to find one of the factors of a given product, when the other is known. According to this definition the divisor multiplied by the quotient must produce anew the dividend; the dividend therefore must contain all the factors both of the divisor and quotient; Whence the quotient is obtained by striking out of the dividend the factors of the divisor. Thus to divide a b c d by a c, we strike out of the dividend the factors a and c of the divisor and obtain b d for the quotient. 2. Let it be required to divide a5 b by a2 b. Decomposing a5 into the two factors a3 and a3, the dividend may be put un der the form a3 a2 b; whence striking out of the dividend the factors a2 and b of the divisor, the quotient will be a3. From this example it appears that in order to find the quotient of two powers of the same letter; From the exponent of the dividend we subtract that of the divisor, the remainder will be the exponent of the quotient. 3. If it be required to divide 72 a b c by 9 b2, we find that 72, the coefficient of the dividend, may be decomposed into the two factors 9 and 8; 67 may also be decomposed into the two factors b5 and 62; the dividend therefore may be put under the form 9X8 a b5 b2 c; whence, suppressing 9 and 62, the factors of the divisor, we have 8 a b5 c for the quotient. From what has been said we have the following rule for the division of simple quantities, viz. 1o. divide the coefficient of the dividend by the coefficient of the divisor; 2°. suppress in the dividend the letters, which are common to it and the divisor, when they have the same exponent, and when the exponent is not the same, subtract the exponent of the divisor from that of the dividend and the remainder will be the exponent to be affixed to the letter in the quotient ; 3°. write in the quotient the letters of the dividend, which are not in the divisor. EXAMPLES. Ans. 4 a2 b c d. Ans. 5 a2 b3 c d. in order that the 1. To divide 48 a3 b3 c2 d by 12 a b2 c 2. To divide 150 a5 b8 c d3 by 30 a3 b5 d2 43. From the preceding rule, it is evident, division may be possible, 1o. that the coefficient of the divisor should exactly divide the coefficient of the dividend; 2°. the exponent of a letter in the divisor should not exceed the exponent of the same letter in the dividend; 3°. that there should be no letter in the divisor, which is not found in the dividend. When these conditions do not exist, the division can only be indicated by the usual sign. If it be required, for example, to divide 12 a2 b by 9 c d, the division, it is easy to see, cannot be performed, we therefore express the quotient by writing the divisor under the dividend in the form of a fraction, 12 a2 b 44. The expression is called an algebraic fraction. Fractions of this species may be simplified, in the same manner as those of arithmetic, by striking out the factors, which are common to both terms, or which is the same thing, by dividing both terms by the factors, which are common to them. Let it be required, for example, to divide 48 a3 b2 c d3 by 36 a2 b3 c2 de; from what has been said, the most simple expression for the quotient will be 4 a d2 In like manner a2 b divided by 5 a3 b gives tient. 1 for the quo5 a 45. It sometimes happens, that the exponent of a letter is the same both in the divisor and dividend. The rule for obtaining the exponents of the letters of the quotient, art. 42, being applied to a case of this kind, will give zero for the ex a2 ponent of the letter in the quotient. Thus, according to a2 a2, a2 this rule gives a° for a quotient; but it is evident, is equal to unity, the expression a° may therefore be considered as equivalent to unity. In general, a letter with zero for an exponent is to be regarded as a symbol equivalent to unity. This symbol, it is evident, will produce no effect upon the value of the expression, in which it appears as a factor, since it signifies nothing but unity. Its only use is to preserve in the work the trace of a letter, which formed a part of the question proposed, but which would otherwise disappear by the effect of division. Thus, if it be required to divide 24 a3 b2 by 8 2 b2, the quotient from what has been said may be put under the form 3 a b°. The symbol b° indicates that the letter b enters o times as a factor in this result, or in other words that it does not enter into it as a factor, but at the same time it serves to show that this letter belonged as a factor to the quantities, from which the result 3 a is obtained by division. 46. We pass next to the division of polynomials. Since the divisor multiplied by the quotient should produce anew the dividend, it is evident, that the dividend must contain all the partial products arising from the multiplication of each term of the divisor by each term of the quotient. This being the case, it is easy to see, that if we can find any one of these partial products in the dividend, and the particular term of the divisor upon which it depends is known, by dividing this term in the dividend by the known term of the divisor, we shall obtain a term of the quotient sought. by Let it be required to divide 50 a3 b2-41 a1 b + 20 a5 + 10 a b4 - 33 a2 b3 It is evident from what has been said, art. 40, that the term a5, being affected with the highest exponent of the letter a in the dividend, must have been formed without any reduction from the multiplication of 5 a3, the term affected with the highest exponent of the letter a in the divisor, by the term affected with the highest exponent of the same letter in the quotient, that is, the term 20 a5 of the dividend is the product of 5 a3 of the divisor by a term of the quotient; whence, dividing 20 a5 by 5 a3, we obtain 4 a2 one of the terms of the quotient sought. Multiplying the divisor by 4 a2, we produce anew all the terms of the dividend, which depend upon 4 a2, viz. 20 a3 b2 - 16 a1 b+ 20 a5; subtracting these from the a3 dividend, the remainder must contain all the partial products arising from the multiplication of each one of the remaining terms of the quotient by each term of the divisor. Regarding this remainder as a new dividend, it is evident, from what has been said, that the term 25 a4 b must have arisen from the multiplication of 5 a3 by the term affected with the highest exponent of the letter a in the remaining terms of the quotient sought; whence, dividing — 25 a4 b by 5 a3, we shall be sure to obtain a new term of the quotient. With regard to the sign, which should be prefixed to this term of the quotient, it is evident, that it should be the sign -; since, from the nature of multiplication, the divisor hav ing the sign +, the quotient must have the sign that their product may produce anew the dividend in order - 25 a4 b. Performing the operation therefore, we have 5 a b for another term of the quotient sought. Multiplying the divisor by this term of the quotient, we obtain all the terms of the dividend, which depend upon - 5 a b, viz. -25 a2b320 a3 b2-25 a4 b, subtracting these from 30 a3 62-25 a1 b + 10 a b1 — 33 a2 b3, the remainder 10 a3 b2+10 a b1. 8 a2 b3, will contain all the partial products arising from the multiplication of each one of the remaining terms of the quotient sought by each term of the divisor; whence, for the same reasons as before, dividing 10 a3 b2 by 5 a3, we have 2 b2 for a new term of the quotient; multiplying the divisor by this term and subtracting as before, nothing remains; the division is therefore exact, and we have for the quotient sought 4 a2-5ab2b2. 47. In the course of reasoning pursued above, we have been obliged to seek in each of the partial operations the term in the dividend, affected with the highest exponent of one of the letters, in order to divide it by the term of the divisor, affected with the highest exponent of the same letter. We avoid this research by arranging the dividend and divisor with reference to the same letter; for, by means of this preparation, the first term at the left of the dividend and the first term at the left of the divisor will, in each of the partial operations, be the two terms which must be divided, one by the other, in order to obtain a term of the quotient. The following is a table of the calculations in the preceding example, the dividend and divisor being arranged with reference to the letter a, and placed one by the side of the other as in arithmetic. 20a5-41 a+b+50 a3 62-33 a2b3+ 10 ab45a3-4a2b+5 a ba 20 a5-16 a4b+20 a3 b2 4a-5ab+262 |