or 98. The division of algebraic quantities can be sometimes facilitated by decomposing, at sight, a quantity into its factors; thus, in the above example, the divisor forms the three last terms of the dividend, it is only necessary to seek if it be a factor of the three first; but those have visibly for a common factor 4a", for 826 -4a3b2 + 4a=4a3 x (223-62+1). By this observation, the dividend will become 4a(2a3 --b? +1)+2a3—62 +1, (2a® — 62 +1)*(4a3 +1): therefore the division is immediately effected, by suppressing the factor 2a3-62+1 equal to the divisor, and the quotient will be 4a'+i. Experience, in algebraic calculations, will suggest a great many remarks of this kind, by which the operations can be frequently abridged. 99. It sometimes happens that, in arranging the dividend and the divisor according to the same letter, there occur several terms in which this letter has the same exponent: In this case, it is necessary to range in the same column those terms, observing to order them according to another letter, common to the two quantities. Ex. 10. Divide-a+62 +62c4-ac-Q® +2a*c +66 +2b4c2 +02 64 by a? ---12-c. Ordering the dividend according to the letter a, we will place in the same column the terms --a*b2 and +2a*co, in another the terms + a2b4and -a 64; finally in the last column the three terms +66, +264c", +°c, ordering them according to the exponents of the letter b; then the quantities, so arranged, will stand thus : Dividend. Divisor. - - a*b* a*b* + b a2 -02 ---C +-2a4c-ac4 +2b4ca + b3c4 Quotient. +26 --a4b2 -a* - 2ab 6 +a-h Ex. 11. Divide ax' - (b + ac)23 f (c+bc ta)ą“ --(co+b)x+c by axa-bx+c. +6x3 tacx 3 - bcxa Dividend. -cx 2 — C#+1 tocax 100. The following practical examples may be wrought according to either of the methods pointed out, (Art. 93, 97); but in complicated cases, the tatter should be preferred: See Example 10. Ex. 12. Divide x-** 3--2 + 2x-1 by 22 fox-1. Ans, 3+*+1. Ex. 13. Divide a--- 5a' x + 10a. -10a?x?4 5ax4 --- • by a’ — 3ao . +3ax2 -13, Ans. aa-2ax +r?. Ex. 14. Divide 2x3-193° +261-16 by r-3. Ans. 2x2-3x + 2. Es. 15. Divide 48y-76aya -64a'y +105a3 by ly-3a. Ans. 24yo —2ay-35a2. Ex. 16. Divide a-be by a-b. Ans. atb. Ex. 17. Divide a4- -x* by a? —2. Ans. a tor?. Es. 18. Divide a--b* by a3 + 2a2b+ 2ab +53. Ans. a3-2a2b + 2ab2-63. Ex. 19. Divide at +12 +-64 by a'-ab+ba. Ans. a' tab+89. Ex. 20. Divide 2546 -_24_2x3—8x2 by 5x 31.22. Ans. 5.03 +402 + 3x +2. Ex. 21. Divide a® 4ab to 463 +ca by a +26. c? Ans. a +26+ a+264 Ex. 22. Divide 844 - 2a3b13a2b2 -- 3ab3 by 1a2 -f- 5ab +62. Ans. 2a2 - 3ab. Ex. 23. Divide 20a5 -41a*b +50a3b" - 45ab$-25ab4-665 by 4a--5ab+262. Ans. 5a3-4a2b+-5ab-312. Ex. 24. Divide at t 8a x + 24ao x2 + 32ax 34 16x* by a+2x. Ans. a3 +6a2x+12ax? + 5.03. Ex. 25. Divide x4--0--6).3 +(p-ab+3)x2 + bp--3a)x+3p by x-ax+p. Ans. x2 +60 +- 3. Ex. 26. Divide axu-(a+b)xo +62 by ar--. Ans. x2-ax-h. Ex. 27. Divide ye t-aoy4 +64ya-ao-2bayt*{*y2 - 204b2--a2b4 by y4 + 2a”ya tat--baya +a2b2. Ans. y -a-62 Ex. 28. Divide 9-46x5 +95x2 +150x by a' ----4.5-5. Ans. 9.04 - 10.3 +522 -30.. Ex. 29. Divide 6a* +9a?-15a by 3a2-3a. Ans. 2a2 +-2a +5. Ex. 30. Divide 2a-13a3b + 31a2b--38ab3 + 2464 by 2a--3ab+46. Ans. a* -5ab +66. SV. Some General Theorems, Observations, &c. 101. Newton calls Algebra Universal Arithmetic, This denomination, says LAGRANGE, in his Traité de la Résolution des Equations numériques, is exact in some respects; but it does not make sufficiently known the real difference between Arithmetic and Algebra. Algebra differs from Arithmetic chiefly in this ; that in the latter, every figure has a determinate and individuale peculiar to itself; whereas the algebraic characters being general, or independent of any particular or partial signification, represent all sorts of numbers, or quantities, according to the nature of the question to which they are applied. Hence, when any of the operations of addition, subtraction, &c., are to be made uporumbers, or other magnitudes, which are represunted by the letters, a, b, c, &c., it is obvious that the results so obtained will be general; and that any particular case, of a similar kind, may be readily derived from them, by barely substituting for every letter its real numeral value, and then computing the amount accordingly. Another advantage, also, which arises from this general mode of notation, is, that while the figures employed in Arithmetic disappear in the course of the operation, the characters used in Algebra always retain their original form, so as to show the dependence they have upon each other in every part of the process; which circumstance, together with |