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gle dimension only, I divide the same into the parts 2+2ač, and -bx2ab, which, by inspection, appear to be equal to (x+2a) XX, and (2+2a) X-b. Therefore 2+2a is a divisor to both the parts, and likewise to the whole, expressed by (x+2a) X (2—6); so that one of these two factors, if the fraction given can be reduced to lower terms, must also measure the numerator; but the former will be found to succeed, the quotient coming out exactly 2-ax+bxab; whence the fraction itself is reduced to zo -ax+bxab

; which is not reducible further, by 2-b, since X-6 the division does not terminate without a remainder, as upon trial will be found. az_63)a^—~*(a (See p. 36, ex. 2, last expression.)

x

ataba Divide this +ab8–74 by B*, we have ab)a?–6(ao+ab+8*

aal Ans. ab.

tab

a2tab? a_axmaxi +2)a—2*(at-x

tab2_83
a'a'x-axtaxt

tab2_83
axtar axx

a x-a%x?-ax3+x* Dividing this remainder, 2aRx?_224 by 2x", or leaving out 24", which is found in each term of the remainder, the next divisor is

a?-?)

ax-ax+x(@ma
a ax

—a2x+ Hence a_22 is therefore —aʻitz, the greatest common measure of the two quantities, and if they be respectively divided by it, the fraction is reduced to its lowest terms.

This quantity, 2xé, found in every term of one of the divisors, 2aRx?_2&", but not in every term of the dividend, azaʻx-ax+x, must be left out; otherwise the quotient will be fractional, which is contrary to the supposition made in the rule : and by omitting this part, 2x*, no common measure of the divisor and dividend is left out; because, by the supposition, no part of 2x2 is found in all the terms of the dividend.

ao fo x a2 xt-a 3.2?_22x+39 5. Reduce

each at_74Z*_' 300_a 28! **_112° +39x45

atta%b2-434 1 to its lowest terms. Ans.

32-13

and a2+62 ata2 8x-+-16

atx

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a

a

2a2)

xa (222 a 34)-1(a?
xa2x

a2a2bt This divided taʻzi-a* by a? +ab9_6 This, divided gives -a)x-a (1

by 5*, gives a—b) — (a?
b

?a?
x a2

aa222 Hence _a is the simple This, again divided, a'b2 by divisor.

b", gives a ?-_*)a_bo(1 x2a2 1

a_02 -a2 x a2 Therefore a_ is the simple divisor.

a–68 a**a*b**** Hence ab)

-84 ato 48x4 +16x4-15 20x4+x-1 6. Reduce

and

each of 24.x*_2224 +5 25x* +50°—2–1 these fractions to its lowest terms.

The greatest common measure is 12x4—5 and 524—1, and the reduced fractions are

4r +3 4x+1

and

5x+x-1° 6a9+15ab44a904—10a%bc% al_cd-acto 7. Reduce

9a%

b27abc-6abc +18bc' 4a'd-4acd2ac4-20 each to its lowest terms. 924+2x+4x+m2+1

a_19 8. Reduce

and each 15x4–2x+10x2+2'>0*c*' to its lowest terms. (For ans. to 7 and 8, see p. 59.)

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ADDITION OF FRACTIONS IN ALGEBRA.

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and

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7

, or

RULE. Reduce the fractions to a common denominator, add the numerators together, and under this sum place the common denominator; the result will be the sum of the fractions required. 3x

4.2 1. Add i or

and together. 2a

4

5 5 3xx5+2axa 15x+2ax 3X4XX 2XXX4 XX2X3

+ 2a x5=10a

+ 10a 2X3X4 2.3.4

2.3.4 13x 4x X5+7(242) 20x+72-14 27%-14

;
12
5X7

35

35
2x
8x

Зr

a 2. Add 2a, 3a+ and am

; or 2a+

and together.

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5

Ilu

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ax

áX

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2.1 8x 184-40

22x 2a+3ata-ba, + 5 9 5x9

45' 2x 52

1+22 3. Add and OT 152 and

to 5

or and
7
8

oto
gether.
Žx.7+5.5__14x+25x

392 152.8+1+2x__120x+1+22 5.7 35 35

1x8

8 122x+1 ax(b+c)tax(6—c) abxtacrtabxacx

8 fb-c) X(b+c) 02-=b+c) (c) 2abx

• Ans.

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مه

or

or

35'

Or

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5. Add 75

a 2a

or

and

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(di.

12x 32 5y

10x=9

Зу 4. Add

3x-5 and or and

and

to7 5 3

8

7 gether. 12x-5+7.32 60x+212 81% 5y:8+3+3y 40y+9y 7.5 35

3.8

24 494 (102—9) X7+(3x~5) X8 702—63+24.c—40 24 87-8X7=56

56 946_-103

Ans. 56

56 2x+3 3x-1 4x and

together. 4a 5

7 ax36 x 4a=12ab

12a+b+8a+b+1568 20a’b+1563 2aX 6X4=8ab

12ab?

12ab2 56 X 6X36=1563

20a? +62 6X36X4a=12ab' viding by 6)

Ans.

12ab (2x+3) X2X7 28x' +421

28x?+42x+1052~-35+40x2 (3.2-1) X5 X7 =105-35

70% 4x X 5 X 215 4028

682*+14735

Ans. 5X24X7=70.com. den.

70x

2x–5 6. Add and

and

and 23

3

2x together.

2x2 47_72-3
or
or

as required.
w__9
(a+b) (a+b)=a+2ab+ 2 a’ + 2ab +på fod - 2ab +7
(b) (a−b) = 2*-_2ab+b

a2p? (ckb) (a+b) a

2a2+262

Ans.

at6

مه

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ato

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or

a

a

or

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and

z(x+3)+2(2-3) **+-3c+x_32
(x-3)
(x+3)

2_9
(2x–5)22—3(2-1)

2. X360
2+3

ato 32-7 7. Add and

and
x+3
2-3 atbexd

8 42 3x 26

and d together. 7 2' 3c (243) (243)+(2+3) (x+3) 2*+6x+9+x2–6x+9 (2+3) (2-3)=-9

2—9 2x+18

or

وفر

new nums.

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(3«—7)7+8X4z=212—49+32x 53.X-49

Ans. to the last. 8X7356

56 3x X3=9 cx

Hence the fractions reHere 26 x 2a-4 ab

quired are dx 2a X3c=6acd

9că 4ab 6acd

and 1X30X2=bac, com. denom. вас вас бас Here the three fractions when reduced are 3 23

4x 5a +4x) and at 4 3

5

5 3X3X545

... The new fractions are 2x X4X5=40x

nums.

45 40% 15a +42) X4X3=60a-+48x

60a+48.3

and 60' 60

60 4X3X5=60, com. denom. 2x7 X (243)

7ах

Hence 7a7ax 31X2 X(ar) = bax— 62

140-14.c' (22)X2X7 = 14a+14.x am

bał_622

14a+143

and 2x(a—x) X7= 142–14x common 140-142 140-142 denominator.

as required. + 1

and 3.5

to a common denominator.

1+2 2X5X (+)

5x+52 3x(x+1)*(1+x)=3(x+2x+1)=3x° +62+3 numerators. 3x5x(1-2)= 15(1-2) =15_15x 3X5X(1+x)=15+15%, the common denominator.

50+50 3x® +6x+3 Hence the new fractions are

and

15+15x 15+152 18x_-150

as tequired. 15415x

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nums.

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8. Reduce *1 3x1271111z=1=? .

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and

42 2x
а аа

20-3 5a 9. Add

or
or

and
За 56

each set of fractions 3' 4' 5 4

8' together.

at3

20-5 3a* 10. Add 2a+

ato
to 4a+
or 6a,

together.
5
4

36 5a 6a 3a+2 11. Add

and 4'5'

or 2a, 7

8 3a

5a 10a 4a 12. Add sat. and 2a4

8' 9

46

За,

a

and 3to each together. and a together.

or

SUBTRACTION OF ALGEBRAIC FRACTIONS.

or

or

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2+1

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51. RULE. Reduce the fractions to a common denominator, if required; then place the difference of their numerators over the common denominator, and it will be the difference required. 43 3x+1 4x+2 2-3 1

1 1. From take

-take

take 5

3

Зх

XY xty 4x 3x+1

4x(x+1)-5(3x+1) 4r +4.x---15245 5 2+1 5(x+1)

5x+5 44_11245 (4x+2)3x--(2x-3)3 122? +6x46x+9 5x+5 3x3x=9x

9x 12x +9 4x +3 1

1 1x(x+y)-1X(2-y) 92 Зх

+y (2-y)(x+y)= 2y

Ans. to the last. xy' ata

atx take

take a
alax)'
a+2

a2—2axt x a’+2ax+2? Reduced and

and
a(a+x) alam) ala-2) ala?-_-2)
_2ax+x a+2axtda

42€ at

Ans. alam) a(a-24) ala:- **) 2d. Reduced are a ta

a2_2arta ai+2axt? _20°+2x2

(am a2x

ax 3:1

2x - 36 3. From at

or 4x+

take 3.x~ 20

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2. From a talat 2)

or at ata

A

are

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