24.

25.

Divide 3x2 + 4abx3 — 6a2b3x - 4a3b3 by x+2ab.

Divide the product of x3-3x2 + 3x − 1, x2 - 2x + 1 and x-1 by x1- 4x3 + 6x2 − 4x + 1.

26. Divide 6a* − a3b + 2a2b2 + 13ab3 + 4b by 2a2 - 3ab + 4b2. 27. Divide x + y3 + 3xy - 1 by x+y-1.

29. Divide 2a7b-5ab2 — 11a5b3 + 5a*b* — 26a3b5 +7a2bo — 12ab7

by a* - 4a3b + a2b2 — 3ab3.

30. Divide a b2+2abc2 — a2c2 — b2c2 by ab + ac-bc.

31. Divide the product of a+b−c, a−b+c, and b+c− a by a2-b2-c2+2bc.

32. Divide (a + b + c) (ab + bc + ca) – abc by a + b.

33. Divide (a− bc)3 + 8b3c3 by a2 + bc.

34. Divide b(x − a3) + ax (x2 — a2) + a3 (x − a) by (a + b)(x − a).

35. Divide xy3 + 2y3z — xyz + xyz2 — x3y — 2yz3 + x3z − xz3 by Y+z―x.

36. Divide a2(b + c ) − b2 (a + c) + c2 (a + b) + abc by a−b+c. 37. Divide (a - b) x3 + (b3 — a3)x+ab (a2—b3) by (a−b)x+a2 — b2. 38. Divide ax2- ab2 + b2x-x3 by (x+b) (a-x).

39. Divide (b- c) a3 + (c − a)b3 + (a−b) c3 by a3 — ab ac + bc. 40. Divide (ax + by)2 + (ay − bx)2 +c2x2 + c3y2 by x2 + y2. 41. Divide a b-bx2 + a2x - x3 by (x+b) (α − x). 42. Resolve a2 - b3 — c2 + d3 — 2 (ad – be) into two factors. 43. Divide b(+ a3) +ax (x2 — a2)+a3 (x + a) by (a + b) (x + a). 44. Shew that (x2 − xy + y2)3 + (x2 + xy + y2)3 is divisible by 2x2 + 2y3.

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B = ca- q2, C = ab − r3, P = qr — ap,

45. Shew that (x + y) − x2 — y1 is divisible by (x2 + xy + y2)3. 46. If A = bc - p2, Q=rp-bq, and R=pq- cr, find the value of

BC-P2 CA-Q1

48. Resolve 4a2b2 — (a2 + b2 — c2)2 into four factors.

49. Resolve 4 (ad + bc)2 — (a2 — b2 — c2 + d2)2 into four factors.

50. Shew that (ay — bx)2 + (bz — cy)2+(cx − az)2 + (ax + by + cz)3 livisible by a2 + b2 + c2 and by x2 + y2 + z2.

V. NEGATIVE QUANTITIES.

72. In Algebra we are sometimes led to a subtraction ich cannot be performed because the number which should subtracted is greater than that from which it is required to subtracted. For instance, we have the following relation: (b + c) = a − b − c; suppose that a = 7, b = 7 and c = 3 so that c=10. Now the relation a- (b + c) = a − b − c tacitly supes b+c to be less than a; if we were to neglect this supposifor a moment we should have 7-10-7-7-3; and as 7-7 ero we might finally write 7-10 - - 3.

73. In writing such an equation as 7-10: - 3 we may be erstood to make the following statement: "it is impossible to 10 from 7, but if 7 be taken from 10 the remainder is 3."

74. It might at first sight seem to the student unlikely that an expression as 7-10 should occur in practice; or that if d occur it would only arise either from a mistake which could stantly corrected, or from an operation being proposed which as obviously impossible to perform, and which must therefore bandoned. As he proceeds in the subject the student will however that such expressions occur frequently; it might en that a-b appeared at the commencement of a long investin, and that it was not easy to decide at once whether a were er or less than b. Now the object of the present chapter is ew that in such a case we may proceed on the supposition a is greater than b, and that if it should finally appear that a is

75. Let us consider an illustration. Suppose a merchant to gain in one year a certain number of pounds and to lose a certain number of pounds in the following year, what change has taken place in his capital? Let a denote the number of pounds gained in the first year, and b the number of pounds lost in the second. Then if a is greater than 6 the capital of the merchant has been increased by a-b pounds. If however b is greater than a the capital has been diminished by b-a pounds. In this latter case a -b is the indication of what would be pronounced in Arithmetic to be an impossible subtraction; but yet in Algebra it is found convenient to retain a − b as indicating the change of the capital, which we may do by means of an appropriate system of interpretation. Thus, for example, if a = 400 and b=500 the merchant's capital has suffered a diminution of 100 pounds; the algebraist indicates this in symbols, thus

400-500-100,

and he may turn his symbols into words by saying that the merchant's capital has been increased by 100 pounds. This language is indeed far removed from the language of ordinary life, but if the algebraist understands it and uses it consistently and logically his deductions from it will be sound.

76. There are numerous instances like the preceding in which it is convenient for us to be able to represent not only the magnitude but also what may be called the quality or affection of the things about which we may be reasoning. In the preceding case a sum of money may be gained or it may be lost; in a question of chronology we may have to distinguish a date before a given epoch from a date after that epoch; in a question of position we may have to distinguish a distance measured to the north of a certain starting-point from a distance measured to the south of it; and so on. These pairs of related magnitudes the algebraist distinguishes by means of the signs + and -. Thus if, as in the preceding Article, the things to be distinguished are gain and loss, he may denote by 100 or by + 100 a gain, and then he will denote by -100 a loss of the same extent. Or he may denote a loss by 100

or by + 100, and then he will denote by - 100 a gain of the same extent. There are two points to be noticed; first, that when no sign is used is to be understood; secondly, the sign + may be ascribed to either of the two related magnitudes, and then the sign - will throughout the investigation in hand belong to the other magnitude.

77. In Arithmetic then we are concerned only with the numbers represented by the symbols 1, 2, 3, &c., and intermediate fractions. In Algebra, besides these, we consider another set of symbols - 1, 2, 3, &c., and intermediate fractions. Symbols preceded by the sign - are called negative quantities, and symbols preceded by the sign + are called positive quantities. Symbols without a sign prefixed are considered to have + prefixed.

The absolute value of any quantity is the number represented by this quantity taken independently of the sign which precedes the number,

78. In the preceding articles we have given rules for the Addition, Subtraction, Multiplication, and Division of algebraical expressions. Those rules were based on arithmetical notions and were proved to be true so long as the expressions represented such things as Arithmetic considers, that is positive quantities. Thus, when we introduced such an expression as a-b we supposed both a and b to be positive quantities and a to be greater than b. But as we wish hereafter to include negative quantities among the objects of our reasoning it becomes necessary to recur to the consideration of these primary operations. Now it is found convenient that the laws of the fundamental operations should be the same whether the symbols denote positive or negative quantities, and we shall therefore secure this convenience by means of suitable definitions. For it must be observed that we have a power over the definitions; for example, multiplication of positive quantities is defined in Arithmetic, and we should naturally retain that definition; but multiplication of negative quantities, or of a positive and a negative quantity has not hitherto been defined; the terms are

It is therefore in our power

at present destitute of meaning. to define them as we please provided we always adhere to our definition.

79. The student will remember that he is not in a position to judge of the convenience which we have intimated will follow from our keeping the fundamental laws of algebraical operation permanent, and giving a wider meaning to such common words as addition and multiplication in order to insure this permanence. He must at present confine himself to watching the accuracy of the deductions drawn from the definitions. As he proceeds he will see that Algebra gains largely in power and utility by the introduction of negative quantities and by the extension of the meaning of the fundamental operations. And he will find that although the symbols + and - are used apparently for two purposes, namely, according to the definitions in Arts. 3 and 4, and according to the convention in Art. 76, no contradiction nor confusion will ultimately arise from this circumstance.

80. Two quantities are said to be equal and may be connected by the sign = when they have the same numerical value and have the same sign. Thus they may have the same absolute value and yet not be equal; for example, 7 and -7 are of the same absolute value but they are not to be called equal.

81. In Arithmetic the object of addition is to find a number which alone is equal to the units and fractions contained in certain other numbers. This notion is not applicable to negative quantities; that is, we have as yet no meaning for the phrase "add - 3 to 5," or "add - 3 to 5." We shall therefore give a meaning to the word add in such cases, and the meaning we propose is determined by the following rules. To add two quantities of the same sign add the absolute values of the quantities and place the sign of the quantities before the sum. To add two quantities of unlike signs, subtract the less absolute value from the greater, and place before the remainder the sign of that quantity which has the greater absolute value.