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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 8a6 – 4a3b2 + 4a=4a3 x (203 --- 62 +1). By this observation, the dividend will become
4a3(2a3-62 +1)+2a3-62 +1,
(2a-6+1) x (40° +1): therefore the division is immediately effected, by suppressing the factor 293–62 +1 equal to the di visor, and the quotient will be 4a3 +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 -atba+%C4 --ac4-08 +2ac -+-66 +264c2 ta'b4 by a---12-c.
Ordering the dividend according to the letter a, we will place in the same column the terms --a*b2 and +-2a4c?, in another the terms ta2b4anda? C'; finally in the last column the three terms +69, +264c”, +-6*c*, ordering them according to the exponents of the letter b; then the quantities, so arranged, will stand thus :
Divisor. -a6 - a'ba ta'b* to be a2-02-02 +-2a4c-ac + 2bc2
+ 62c Quotient. tal-Q*72
-a* - 2a2b -6* -a4c2
Ex. 11. Divide ax* -- (6 + ac)x' + (c+bc+a)x®
b x3 (" (c +b)x+c by axa-bx+c.
x 2 — Cx+1
100. The following practical examples may be wrought according to either of the methods pointed out, (Art. 93, 97); but in complicated cases, the Jatter should be preferred: See Example 10.
Ex. 12. Divide - + + x3---x+ 2x-1 by .ca 4x-1,
Ex. 13. Divide a --5a4x + 10a3_-10aox® + , 5
+ 5ax4-25 by a– 3aR - +3ax: -23,
Ans. aa-2ax +. Ex. 14. Divide 233-193+26x-16 by x-3.
Ans. 2002-3x +2. Ex. 15. Divide 48y3-76aya -64aoy+105a3 by y-3a.
Ans. 24y: ---Qay-35a2. Ex. 16. Divide a--ba by a-b. Ans. a+b. Ex. 17. Divide a4. --x* by a'—2.
Ans. a tox?. Ex. 18. Divide ale by a3 + 2a2b+ 2ab +53.
Ans.a-2a2b + 2ab2_63. Ex. 19. Divide at ta’/2 +-64 by aa-ab+62.
Ans. a' +ab+. Ex. 20. Divide 25x6 ---24--2x38x2 by 5x 31x2.
Ans. 5.03 +4.co +30 +2. Ex. 21. Divide aa -f-4ab + 463 +.c2 by a +2b.
ca Ans. a + 2b to
a+26 Ex. 22. Divide 894 -_-203b-13a2b2 --3ab3 by 1a2 -4- 5ab +62.
Ans. 2a2 -- 3ab. Ex. 23. Divide 20a5-41ab +-50a'b? -45abo •4-25ab4-665 by 4a2--5ab+262.
Ans. 5a-4a2b+-5ab-362. Ex. 24. Divide a4 + 8a3x + 24a2x2 + 32ar34 16.04 by a + 2x.
Ans. a3 +6ax+12ax? +8.03. Ex. 25. Divide x4---((-6).3+(p-ab+3)x2 + (bp--3a)x+ 3p by xa--ax+p. Ans. x2 + b +-3, Ex. 26. Divide ax3---(aa+b)x2 +62 by ax --b.
Ans. 2 -ax-b. Ex. 27. Divide y ta`yt +boya-6--2b2y* --#*ya-2a-b2--a*b* by y' + 2a”ya tal--baya +a2b2.
Ans. y -a? -62 Ex. 28. Divide 9.7-4605 +-95x2 +150x by a ---4.0 --- 5.
Ans. 924 - 10.3 +52 -30.4.
Ex. 29. Divide 6a+ +9a2-15a by 322 3a.
Ans. 2a + 2a +5. Ex. 30. Divide 2a4 -13a3b + 31a2b2-38abs + 2464 by 202 -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 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