3 Ex. 28. Multiply 302 — 2xy +5 by 2+- 2xy--3. Ans. 30° +4x'y-4x2 (1+y)+16xy--15. Ex. 29. Multiply a? +-3aRb+3ab2 +63 by q3 3a2b +- 3ab2 -63. Ans. ao — 3a4b2 + 34264-66. Ex. 30. Multiply 5a3 -4a2b+ 5ab2 – 363 by 4a2 --5ab+262. Ans. 20a5-41a1b+50a3b2-45a213 +25ab4-665. § IV. Division of Algebraic Quantities. 80. In the Division of algebraic quantities, the same circumstances are to be taken into consideration as in their multiplication, and consequently the following propositions must be observed. 81. If the sign of the divisor and dividend be like, the sign of the quotient will be +; if unlike, the sign of the quotient will be The reason of this proposition follows immediately from multiplication : +ab Thus, if tax+b=+ab; therefore + = :76: ta -ab tax-b=-- ab; ta - ab -ax+b=-ab; +b: - tab -ax-b=tab; -b 82. If the given quantities have coefficients, the com efficient of the quotient will be equal to the coefficient of the dividend divided by that of the divisor. 4ab Thus, 4ab-;-2b, or -=2a. 26 For, by the nature of division, the product of the quotient, multiplied by the divisor, is equal to the dividend; but the coefficient of a product is equal to the product of the coefficients of the fac 4 ba tors (Art. 70). Therefore, 4ab-1-2=X =2a. 2 b 83. That the letters of the quotient are those of the dividend not common to the divisor, when all the letters of the divisor are common to the dividend : for example, the product abc, divided by ab, gives e for the quotient, because the product of ab by c is abc. 34. But when the divisor comprehends other letters, not common to the dividend, then the division can only be indicated, and the quotient written in the form of a fraction, of which the numerator is the product of all the letters of the dividend, not common to the divisor, and the denominator all those of the divisor not common to the dividend : thus, abc divided by amb, gives for the quotient ' in observing that we sup 3 m press the common factor ab, in the divisor and dividend without altering the quotient, and the division is reduced to that of S, which admits of no farther reduction without assigning numeral values to c с m and m. 85. If all the terms of a compound quantity be divided by a simple one, the sum of the quotients will be equal to the quotient of the whole compound quantity. ab ac, ad ab+ac+ad =b+c+d. + a a a a a 36. If any power of a quantity be divided by any other power of the same quantity, the exponent of the quotient will be that of the dividend, diminished by the exponent of the divisor. Let us occupy ourselves, in the first place, with the division of two exponentials of the same letter; for instance, m and n being any positive whole an? numbers, so that we can have, m>n, m=n, m<n. It may be necessary to observe that, according to what has been demonstrated (71), with regard to exponentials of the same letter, the letter of the quotient must also be a, and if the unknown exponent of a be designated by x, then at will be the quotient, and from the nature of division, a"=a" Xa*=antx; from which there necessarily results the following equality between the exponents, m=n tox; And as, subtracting n from each of these equal quantities, the two remainders are equal (Art. 49), we shall have, (1). Therefore, in the first case, where m is >n, the exponent of the quotient is ma-n; thus, 5 3 = 15-3=u”, and as raras Also, it may be demonstrated in like manner, that (a + x) = (a + x)2 =(a + x)5—3=(a + x)) ; and (2x+y)” =(22+y) 7-5 =(2x+y). (2x+y)^ In the second case, where m=n, we shall have, am=an Xa*=a" Xar=a'n**; 3 3—1=a. a 5 am From which there results between the exponents the equality, m=mtor, and subtracting m from each of these equals(Art. 49), m-m=x, or x=0....(2); therefore, the exponent of the quotient will be equal to 0, or a*=a', a result which it is necessary to explain. For this purpose, let us resume the division of am by a”, which gives unity for the quo am tient, or =1; and as two quotients, arising from the same division, are necessarily equal; therefore, a=1. Hence, as a may be any quantity whatever, we may conclude that; any quantity raised to the power zero, must be equal to unity, or 1, and that reciprocally unity can be translated into ao. This conclusion takes place whatever may be the value of a; which may also be demonstrated in the following manner. Thus, let ao=y; then, by squaring each member, y ; but ao =y; consequently ao=1. In the third case, where m is less than n ; let n=m+d, d being the excess of n above m; we shall always have, a"=amid Xam=am+d+x. and equalising the exponents, because the preceding equality cannot have place, but under this consideration, m=mt.dtox, or y (3); . d subtracting on+d from both sides, the final result will be d then the quotient is a d.. In order to explain this, let us resume the division of am by a", or by amd=a" Xad; hy suppressing the factor am, which is common to the dividend and divisor, according to what has been demonstrated with regard to the division of letters (Art. 841). 1 we have for the quotient therefore, 1 . (1): ad This transformation is very useful in various analytical operations; in order to see more clearly the meaning of it, we may recollect that atd is the same as a Xa Xa, &c., continued to. d factors; therefore, according to the acceptation and opposition of the signs, and must representa Xa Xa, &c., continued to d factors in the divisor, Hence, according to the results (1), (2), and (3), the proposition is general, when m and n are any whole numbers whatever; thus, au=3-5=a 1 or eme; because the divisor multiplied by the quotient is equal to the dividend, a5 Xa?=25–2=a? the dividend, andox Xasa -=a5-2=a=the divi a? 3 1 a 5 1 dend, therefore q-2 In like manner it may be a 1 a- a4 -3 > a3 shown that, at=a<\, &c. But, according to the result (4), in general, =a-, where d may , |