Sidebilder
PDF
ePub

23. a3 + 73 + c3 - 3 abc and (a + b)2 + 2 (a + b)c + c?.

24. (a + b)2 – (c + d)?, (a + c)2 – (6 d)’, (a + d)? - (b - c)?, (c + d)2 (a - b)?, (6 + d)a - (a - c), and (6 + c)2 = (a - d)?

CHAPTER V.

Fractions. 56. It is unnecessary to repeat here the propositions relating to fractions which were proved in Arithmetic, Chap. II. of this work. The student will see that, by substituting general symbols for the particular figures there used, the reasoning will equally hold. We shall work out a few examples to show the method of dealing with them in algebra.

.. 23 - 2 202 + x - 12 Ex. 1. Simplify the fraction

sca + 2 x – 15 By inspection (Art. 30) we see that » – 3 is a factor of numerator and denominator. We have then2ct – 2 x2 + x – 12 _ *2 (oc – 3) + x (oc – 3) + 4 (x – 3) ca + 2 x – 15

(C – 3) (2 + 5) :- (x2 + x + 4) (x – 3) 22 + x + 4 (2 – 3) (x + 5) = x + 5 . Ans.

1

1 Ex. 2. Find the value of 1īta 7

12. 1

1 2 a (a - b) + (a + b) 2 a a + b * a -7 + 62 (a + b) (a - b) a2 + 6% - 2a 2a 1 1 1

az 63 * + 12 = 2 la? - 72 ^ a + ?) Lo (a+ 62) - (a62) 262 - 4 al = 2 &Ta63) (a + b2) = 2d.at - 3* a* - 74

2 a

[ocr errors]

bc

Ex. 3. Find the value of (a - b)(a – c) + T-(6-c)

ас ab + c a) (c 6).

The second denominator has a factor, (6 a), which differs from a factor, (a - b), of the first denominator in sign only. We shall therefore change the sign of the second fraction, and also of its first factor. This will not alter its value.

And, similarly, we find that by changing the signs of each of the factors of the third denominator we shall have them in a form corresponding to factors of the first and second denominators. The sign of the third fraction will not be changed, as the sign of the denominator will, on the whole, be unchanged. The given expression then will stand thusbc ac

a b (a - b) (a c) (a - b) (6 c) * (a – c) (6 c) bc (6 c) ac (a – c) + ab (a - b)

(a - b) (a c) (6 c) bc (6 c) – aʻc + ac + a2b ab? (a - b) (a c) (6 c)

-, or, re-arranging, a(6 c) a (62 – c) + bc (6 - 0

then, dividing nume(a - b) (a c) (6 c) rator and denominator by c Laa (b + c) + bc (a - b) (a – c) (a - b) (a – c)

d
(a b) (a – c) =

ia -o = 1.
Ex. 4. Simplify,
La 4a ́x – 3 ax + 200 L2, 4 aʻc + 3 ax? + 2c3
a + 2C

"
------

The given expressiona(a + ) – 4 ařx + 3 ax2 – 2013 ala x) + 4 aʻoc + 3 axé + 203 a + 2C

a - 20 a+, aʻx – 4 aʻx + 3 ax2 – 203 aaʻx + 4 aʻx + 3 ax? + 23 a + x

a – a

X

X

ai - 3 aʻx + 3 ax? - Qusa+ 3 aʻx + 3 ax? + 203 a +

a - X = (a – x)" x (a + x)* = (a – a) x (a + x)2 _ , - at æ*a - = 1X I -= (a“ – 2). Ex. 5. Divide

1 1
- t

by (a
12 - a2 + b )

a) * (c + b) S Now2 1 1 )

* (x + 5) – (oc – a) X a 2 + b S

(x – a) (x + 6)

[ocr errors]

(a + b) 3

[ocr errors]

(a

+

6).

[ocr errors]

(a + b) · Tc - a) (a + b).

(a - b).

[ocr errors]

i

(a - b) (a + b) ac? + ab (a + b) –

, (oc – a)? (c + 5)2

(ac – a) (x + 6) _ (a + b) (x – a) (x + b) (a - b) (22 + ab)

Ex. XIV. Simplify the following expressions:1 x – 5x + 4 23 - 3x + 2

ca + 2 x – 24' 203 + 4 x – 5°

6 x2 + 29 x + 35 22c3 + 73 - 9 - 14x2 + 39 x + 10' 5 – 32 - 4x + 2

a* 2 a2b2 + 64 24 a3 – 28 ab + 6 ab? - 763 0. a} – 4 ab + 4 ab? 73 – 6 a2 + 11 ab 2162 iac? (yo – ) – 2y (2 y + yz – *) + 49 (y + 2) 2* + 2*Y* +34 *** (y + z)xy(2 y + 3 yz + x) + yd (y+z)' 28 30

1 1 1*
a + 7 a -7 a -7 ā + 6

[ocr errors]
[ocr errors]
[ocr errors]

a b

ab , ab
a + b ā- 7 ab 62 * a® + abi

1 - 2 - 1 2 - 1
4(x - 1) 4(* + 1) – 21a2+ 1);

18 7 x + 1
x + 1 (wc + 2) 5 (+ 2) 5 (ac? + 1)
o 11 11 14
* x + 1 2 + 3 (6 + 1)

2 16 x + 14 16 x – 8 1)*(x - 1)3 72 – 1)2 * 3 (– 1) – 3(x* – x + 1):

[ocr errors]
[ocr errors]

+

+ +

(6 - a) (6 – c) +

) (0 a) (b c). (c a) (c 6)* a

b (a - b) (a - c) + 76 a) (6 c) * (€ – a) (c – 7)*

[merged small][merged small][merged small][merged small][subsumed][ocr errors][subsumed]

16. (a + 2)2 + (6 – c)2 + (a + c)2

2 2 (a + b) (6 c) (a + c) a +c b-c' a + 11 , 1 2 a 1 1 1 2 x 1 Tataca -xaž + xSlatxa - 2 ař + S

[ocr errors]
[ocr errors]

[ocr errors]
[ocr errors]

20. a,

[ocr errors]

20. - t =

ab + bc + ca

21. {C+%*+1},{C+2)+1}-{(+3)+1} *

[ocr errors][merged small][merged small][merged small][merged small][ocr errors][subsumed][subsumed][subsumed][subsumed][ocr errors][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small]

(c y) (a + x) (c y) (a + y). (a + y)**

x + y +1
Y 3

+ 12

[merged small][ocr errors][subsumed][ocr errors][merged small]
« ForrigeFortsett »