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10ab-150x (10ab - 15ax) = 5a, or
30ax_483% (30ax-48x2 - 6x, or
= 5a - 8x.
6x 1. Let 3.x3 +6x2 +3ax - 15x be divided by 3x. 2. Let 3abc+12abr - 9026 be divided by 3ab. 3. Let 40a3b3 +60a2b2—17ab be divided by - ab. 4. Let 15a2 bc – 12acx? + 5ada be divided by --5ac. 5. Let 20ax+150x2 +10ax+ 5a be divided by 5a
When the divisor and dividend are both compound
Set them down in the same inanner as in division of numbers, ranging the terms of each of them so, that the higher powers of one of the letters may stand before the lower.
Then divide the first term of the dividend by the first term of the divisor, and set the result in the quotient, with its proper sign, or simply by itself, if it be affirmative.
This being done, multiply the whole divisor by the term thus found; and, having subtracted the result from the dividend, bring down as many terms to the remainder as are requisite for the next operation, which perform as before ; and so on, till the work is finished, as in common arithmetic.
2x2 - 3axta?)4x4 -9a?r? +6a3x-a*(2x2 +3ax - *
4x4 - 6ax3-72a2x2
Note 1. If the divisor be not exactly contained in the dividend, the quantity that remains after the division is finished, must be placed over the divisor, at the end of the quotient, in the form of a fraction : thus ()
(f) In the case here given, the operation of division may be considered as terminated, when the highest power of the letter, in the first, or leading term of the remainder, by which the pro. cess is regulated, is less than the power of the first term of the divisor ; or when the first term of the divisor is not contained in the first term of the remainder; as the succeeding part of the quotient, after this, instead of being integral, as it ought to be, would necessarily become fractional,
2y* x+y)x+ +y4 (x3 x3y +xyz -Y3+ xatx3y
2. The division of quantities may also be sometimes carried on, ad infinitum, like a decimal fraction ; in which case, a few of the leading terms of the quotient will generally be sufficient to indicate the rest, without its being necessary to continue the operation ; thus,
x? 23 24 a + x)a .. (1
And by a process similar to the above, it may be showa that
Where the law, by which either of these series may be continued at pleasure, is obvious.
EXAMPLES FOR PRACTICE.
1. Let a? 2ax + x2 be divided by a - $. 2. Let x 3 3ax2 + 3a2 x - a3 be divided by x-.. 3. Let a3 + 5a2 x + 5ax2 + x3 be divided by at.. 4. Let 2y3 - 19y2 + 26y-- 17 be divided by y-8. 5. Let x5 +1 be divided by 3+1, and 6-1 by -1.
6. Let 4833 — 760x2 - 64a2 x + 105a3 be divided by 2x - 3a
7. Let 4x4 -9:+6x – 3 be divided by 2x2 + 3x - 1. 8. Let x4 – a2x2 + 2a3x – 24 be divided by <2 - áxtal.
9. Let 6x4 – 96 be divided by 33-6, and a5 + x5 by atx.
10. Let 3205 +243 be divided by 2x+3, and 26-QE by x-a. 11. Let 64 - 3y be divided by b-y, and a' +4a3b+
- , 364 by ü+26.
12. Let x2 +px+q be divided by x+a, and x3-px? + 9-rby x
OF ALGEBRAIC FRACTIONS.
ALGEBRAIC fractions have the same names and rules of operation as numeral fractions in common arithmetic ; and the methods of reducing them, in either of these branches, to their most convenient forms, are as follows :