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 4a3, for 8a — 4a3b2 + 4a3 = 4a3 X (2a3-b2+1). 3 By this observation, the dividend will become 4a3 (2a3 —b2+1)+2a3—b2+1, ΟΙ (2a3-b2+1)x(4a2+1): therefore the division is immediately effected, by suppressing the factor 2a3-b2+1 equal to the divisor, and the quotient will be 4a3 +1. 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 b2+b2c4 — a2c2 — a® +2a1c2 +b+2b1c2+a2b by a2-b2-c2. Ordering the dividend according to the letter a, we will place in the same column the terms-a4b2 and +2ac2, in another the terms +a2band -a2 c4; finally in the last column the three terms +6, +2b1c2, +b2ca, ordering them according to the exponents of the letter b; then the quantities, so arranged, will stand thus: Ex. 11. Divide axa — (b + ac)x3 + (c+bc+a)x* ----(c2+6)x+c by ax2-bx+c. Dividend. Divisor. +bx3 -cx2 ax1 —(b+ac)x3+(c+bc+a)x2 —(c2+b)x+c | ax2—bx+c -αx4 -acx3+(bc+a)x2 —(c2+b)x+c -bcx2 +c2x 100. The following practical examples may be wrought according to either of the methods pointed out, (Art. 93, 97); but in complicated cases, the fatter should be preferred: See Example 10. Ex. 12. Divide x-x+x3-x2+2x-1 by x2 4x-1. Ans. xx3x2-x+1. Ex. 13. Divide a55a4x+10a3x2-10a2x2+ 5ax-x by a3-3a2x+3ax2--x3, Ans. aa-2ax+x2. Aus. 2x2-3x+2. Ex. 14. Divide 2x3-19x+26x-16 by x-8. Ex. 18. Divide a-be by a3+2ab+2ab+53. Ans. a3-2a2b+2ab2 —b3. Ex. 19. Divide a +ab+b by a2-ab+b2. Ans. a2+ab+b2. 3 Ex. 20. Divide 25x-x4-2x3-8x2 by 5x3- 1x2. Ex. 22. Divide 8a42a3b-13a2b2-3ab3 by 4a2+-5ab+b2. Ans. 2a2-3ab. Ex. 23. Divide 20a5-41a1b+50a3b2-45a2b +25ab1-6b5 by 4a2-5ab+262. Ans. 5a3-4a2b+5ab2-3b2. Ex. 24. Divide a4+ Sa3x + 24a2x2 + 32ax3 + 16x* by a+2x, Ans. a3+6a2x+12ax2+Sx3. Ex. 25. Divide x1—(a—b)x3+(p—ab+3)x2+ (bp-3a)x+3p by x2-ax+p. Ans. x+bx+3. Ex. 26. Divide ax3-(a+b)x2+b2 by ax-b. 6 Ans. x2-ax-b. Ex. 27. Divide y°+a2y+b4y2 — a ® — 2b2 y1 ——— · ua y2 — 2 aab2 —-a2ba by y1+2a2y2+a1--b2y2+a2b2. Ans. y2a2b2 Ex. 28. Divide 9x46x595x2+150x by a2 -4x-5. Ans. 9x4-10x3+5x2-30x Ex. 29. Divide 6a4+9a2-15a by 3a2-3a. Ans. 2a2+2a+5. Ex. 30. Divide 2a4-13a3b+31a2b2-38ab3+ 2464 by 2a2-3ab4b3. Ans. a-5ab+6b2. 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 individu 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 upo umbers, or other magnitudes, which are represented 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 |