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Note. If the divisor be not exactly contained in the divi. dend, the quantity which remains after the operation is finished. may be placed over the divisor, like a vulgar fraction, and set down at the end of the quotient as in common arithmetic

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2 x4 a + x) a- 3x* (a3 - a'r + ax? - 33

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2az + .

1. Divide a® + 4ax + 4x? by a +- 23.

Ans. at 23. 2. Divide a? 3u>z +- Jaz?

z3 by a – 2.

Ansa? 3. Divide 1 by 1 ta. Ans. I-a + a2 a3 + &c. 4. Divide 12x 192 by 323 6.

Ans. 4.33 + 8x2 + 16.3 + 32. 5. Divide as 5a46 + 10a362 – 10a2b3 + 5a64 65 by ao – 2ab + 62.

Ans. a3

Ja?b + 3ab2 - 63. 6. Divide 487% 96az? – 64a2z + 150a3 by 2z Sa. 7. Divide 66 -36*x+36°x*- x6 by 63-362x+36x-23, 3. Divide a? x07 by a - X. 9. Divide a3 + 5a2x + 5ax2 + x3 by a + x. 10. Divide a4 + 4a2b2 -3264 by a + 26. 11. Divide 2494 64 by 3a 2b.

ALGEBRAIC FRACTIONS.

ALGEBRAIC FRACTIONs have the same names and rules of operation, as numeral fractions in common arithmetic; as appears in the following Rules and Cases.

CASE

CASE I. To reduce a Mixed Quantity to an Imfiroper Praction. Multiply the integer by the denominator of the fraction, and to the product add the numerator, or connect it with its proper sign, + or -; then the denominator being set under this sum, will give the improper fraction required.

EXAMPLES.

6 1. Reduce 3ş, and a - to improper fractions.

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[ocr errors]
[ocr errors]

and a

a

a2

a2 2. Reduce at

to improper fractions. b

a axb + aa ab + a2 First, at

the Answer.
b
b

b
z? -a?

al

z2 + a2 2a And, a -

the Answer,

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a

3. Reduce 5 to an improper fraction.

3a 4. Reduce 1

to an improper fraction.'

Ans. 34.

3a Ans.

1

X

5. Reduce 2a

3ax + a2

to an improper fraction.

4x 18 6. Reduce 12 +

to an improper fraction. 58

1 36 7. Reduce at

to an improper fraction.

233

3a

8. Reduce 4 + 2x

to an improper fraction,

5a

CASE II. To Reduce an Improper Fraction to a Wholeor Mixed Quantity.

Divide the numerator by the denominator, for the integral part ; and set the remainder, if any over the denominator, for the fractional part; the two joined together will be the mixed quantity required.

EXAMPLES

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EXAMPLES.

16 ab + a? 1. To reduce and to mixed quantities:

3

b First, ist = 16 3 = 5j, the Answer required.

ab + a2 And, -= ab + a2 b=a + Answer. 6

b 2ac 3a? 303 + 4.02 2. To reduce

and

to mixed quantities.

a + 2ъс 3a2

322 First,

3a? C= 2a

Answer.
с
Sax + 4.x
And,

-= 3az + 4xOtx= 3x + - Ans, a + x

a + x 33 2ar 3r2 13. Reduce and

to mixed quantities. 5

322 Ans. 63, and 23

= lac

с

x2

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42 2a?

+

262 4. Reduce

and

to whole or mixed quan2a

a - b tities.

3r2
3y2

2x3 2y3 5. Reduce

and

to whole or mixed x + y quantities.

10a2 40 + 6 6. Reduce

to a mixed quantity. 50

15a3 + 5a2 7. Reduce

to a mixed quantity. 3a3 + 2a2 2a-4

CASE III. To Reduce Fractions to a Common Denominator. MULTIPLY every numerator, separately, by all the denominators except its own, for the new numerators; and all the denominators together, for the common denominator

When the denominators have a common divisor, it will be better, instead of multiplying by the whole denominators, to multiply only by those parts which arise from dividing by the common divisor. And observing also the several rules and directions as in Fractions in the Arithmetic.

EXAMPLES

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a

az

IZ

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с

Here

b

bx Here and and --, by multiplying the terms of the

IZ first fraction by z, and the terms of the 2d by x.

2

6
2. Reduce“, – and to a common denominator.

b
6 abc cx2

623
and

and ---, by multiplying the 6

ber bor бcx terms of the 1st fraction by bc, of the 2d by cx, and of the 3d by br.

2a 36 3. Reduce and to a common denominator. 20

4ac 36% Ans. and

2cx 2cx 2a 3a +26 4. Reduce and

to a common denominator.
6
2c

4ac 3ab + 262
Ans. and
26c

2bc 5a 5. Reduce - and ---, and 4d, to a common denominator. 31 20

10ac 96x 24cda Ans. and

and

6cc 6cx 60% 5 За

3a 6. Reduce and

to fractions having a 6

6 206

18ab 486% + 720 common denominator. Ans. and and

246
246

246
1

2a2 + 62 7. Reduce - and and

to a common denomi. 3 4 nator.

36 2c d 8. Reduce and and to a common denominator. 42? 3a 2a

CASE

365,

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and 2b +

2a2

atb

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