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CASE III.

To reduce a mixed quantity to an improper framan.

,

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Multiply the integral part by the denominator of the fraction, and to the product add the numerator, when it is affirmative, or subtract it when negative ; then the result, placed over the denominator, will give the improper fraction required.

EXAMPLES

с

b
1. Reduce 34 and a to improper fractions.

3X57-2 15+2 17
Here 3 =

Ans.
5

5 5
b
а Хc -6

.b
Anda

Ans. с

a2 2. Reduce x +- and a

to improper frac

ac

с

с

a

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To reduce an improper fraction to a whole or mixed

quantity.

RULE.

Divide the numerator by the denominator, for the integral part, and place the remainder, if any, over the denominator, for the fractional part ; then the two, joined together, with the proper sign between them, will give the mixed quantity required.

EXAMPLES.

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27

axta 1. Reduce and to mixed quantities.

5

27
Here

27--5=57. Ans.
5
axtaa

a2
And

(axtas)--=a+ Ans.

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2. It is required to reduce the fraction

ax-73

to a

whole quantity.

ab - 2a2 3. It is required to reduce the fraction

to a

ab mixed quantity

EXAMPLES.

= 26

or

ab +-62
(ab+62)-26, or

at6
= fa+b=
26

2

10ab15ax (10ab - 15ax) = 5a, or

. 5a

30ax_48x2 (300x-4832) • 6x,

= 50 - 8x.

6x 1. Let 3x3 +682 +3ax – 15x be divided by 3x. 2. Let 3abc+12abr -9a26 be divided by 3ab. 3. Let 40a363 +60a2b2-17ab be divided by - ab. 4. Let 15a2 bc - 12acx? +5ada be divided by --5ac. 5. Let 20ax+15ax2 +- 10ax+ 5a be divided by 5a

1

CASE III.

When the divisor and dividend are both compound

quantities.

RULE.

Set them down in the same manner 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 com-mon arithmetic.

EXAMPLES.

x+y)?? + 2xy +y(x+y

22 +xy

xy+y? xyty

*+xa3+5aastbax +33(a* +-4axt

a3 ta

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92-27 9x -27

2x2 – 30x+a2)4x4 - 9ao? +6asr-a (2x2 +3ax - *

4x4

bax3 +2a2x2

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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 (1)

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(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 process 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 succeedi

part of the quotient, after this, instead of being integral, as it ought to be, would necessarily become fractional,

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