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a-C

+

C

—c):+

= 0.

C-a

+

ac

b

a' +63 +1+3abc (6+c)(c+a)(a+b) 9.

+

atazot

a-67 (6-c)(c-a) (a-6) (6-c)(c-a)(a-6) ab+bc+ca

b-a c-b

a' +63 +63 10.

(a+b)(b+c)(c+ala+66+ccta (a+b)(6+c)(c+a)

a' tab +34 a+b a-6 1 2ab(a? +62) 11.

qo— 76 a— 65 a'+63a-62 a6-66
1 1
1 1

1
1

1 12.

+ +
b-c

-b
(62

-ay?+Ta-6)*

2 (ab): +(6c)+(ca)

2 2 13.

+ (a-bb-c)(c-a)

+

attoo 14. a[b+c-a) b(c+ab), ca+6—c)

+
bc

ab
(a + b)(b+c)(c+a)-(a+b3 ++2abc)

abc
a'(b+c)
b'(c+a)

c*(a+b) 15.

a(b+c)-(a'+bc)+b(c+a)-(+ac)+c(a+b)–(+ab) = 0. (a+d bc (b+d)ac

(c+d)ab 16.

+

= d. (a-b)(a-c) (6a)(6

(c-a)(c-b) ( (a-6-(6-c) (6-0)

(6c)-((-a) (c-a-(a-6) 17. +

= 0.
a' tab-bc- 6? + bc-ca-a? c*+ca-ab-02
18.
ao+62—(2 62+(-a'cta62
+

+6
ab
be

11 1 1
ab

+
19.
abcd-(ab+ad+cd—1) abc-(a+c)
bed-(b+d)

bc-1

(ad+bc)(bc—1)—bc+a) = a to

(bed6-d)(bc-1)
11 1 1 il (a+b+c+d)
20.
+7+e+

abcd
s 1 11 1 1 1 1 11

adbe Srba

+

c)

+

+

са

{
lab—(a+b=0}},{

it to

ca

a

XVII.

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Reduce to their simplest forms:

1

1 1. (a-0}{a_e)

+
-c)' (ba)(6c)*(cb)(ca)

6
2.
(ab)(a–o)+(6–4}(6—c)+(0-a)(0–6).

6-0
(6+C)(c+a)t(eta)(a+b)+(a+b)(6+0)
(6c)(ca)+7c-a)(a–6)+(ab)(6—c)

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bc

ca

5.

+

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6.

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(ble)

7.

+

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8.

9.

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10.

ab (6-c)(a-a)*(oa)(a-bot (a-b)(6-0)

72


(a~b)(a-co) (6a)(b-oft (e-a)(cb)

+

(c-a)* (a-b)
(e-a)(abot (aーb)(6c)*(b)(ca)

it

+
(ab)(a-c)(6-c)(b-a)

a*bab+(c-a)(c-b)
d'(c-b) B(a-c) c0a)
(a+b)(a+c) * (b+c)(6+a)+(x+a)(c+d)
+

+ c
ap-bc

cab
(a+b)(a+c)+(b+c){6+a)+(x+a)(e+)
+

ccd
a+b-

b+c-a c+a-6
(6_c)(c-a)*(x-a)(aーb)*(ab)(bc)

+
atb
b+c

c+a
(ambo)(B_ac) *(ジーac)(_ab)*(d-ab)(a+bc)
(1+ab)(1 + ac)(1 + bc)(1+ab), (1+ac)(1 + be)

+
(ab)(c-a) (a-)(6-c) (6-c)(c-a)
«(a+b)(a+c) - 8(6+c)(6+2)+c(c+a)(c+d)

+
(ab)(a-c) (6-c)(b-a) (c-a)(c-b)
a*(a+b)(a+c) , (b+c)(b+a) , (c+a)(c+b)
+

+
(ab)(a-c) (6-c)(b-a) (c-a)(c-b)

11.

+

12.

+

+

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Reduce to their simplest forms the expressions : a_6

b-c 1.

+
(-a)(x+b)*(x-b)(cec)(x-a)(^ー^)
+
+6

x+a 2.

+
(x-a)(c-b) (c-a)(cc) +(z-b)(cm)
3.
(a~b)(ac)+(c)(6a)+(ea)(c_)
+

st
(x+c(x+c
(x+b)(+c) , (x+a)(x+c), (x+a)(x+b)

(+)+
+

+
(ab)(a-c) (6-a)(b-c) (c-a)(c-b)

(z-b)(arc)(rec)(rea) , (xa)(x-b)
5.
c_c, arra

+ (ab)(a-c) (6-0)(b-a)

-a)* (

(c-a)(c-b) 6. 29 _ (_ra)(y_a) (z-b)(yーb).

+ a(aーb)

6(

64) box-a cal cb) ab(rec) 7.

+

+ (aーb)(a-c) (6_c)(b-a)*(c-a)(c-b) 8. (-a)(ya)(z_a) , (b(yーb)(aーb) (ー)(yー)(ーc) +

+ (ab)(a-c) (6-a)(6-0) (c-b)(c-a)

abx 9.

acy

bcz
+

+
(ax-by)(ar-cz)*(byar)(by-cz)
(byar) (byca) + (res ar) (ccby)

z

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ac

ab

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1

3.

C

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4.

+

XIX.
Determine the aggregates of -
1

1
1.

+
a(ab)(ac)*816-c)(-a)

_co=a}to{ca|0–6)
bc
2.
a(a* —62)(a*—c)+b(b? — a*)(62—(2) +{c* 6°) (c*—a*)
1

1 (6—a)(c—a)(x+a) + (abb

-6)(c-6)(x+6)+(a–c)(6—c)(x+c)*

+
(ab)(a—c)(x-a)*(6-a)(6—c)(x-6)*(-a)(c-6)(x—c)
(ab)a—c)(a+2)(ab)(6—c)(6+2)+(a—c)(6—c)(c+x)"
(ab)(a—c)(6+0)*(c)(6—a)(a+c) (c—a)(0–6)(6+a)
bto-a
cta-b

atb-c
(6+c)(
ca)(a—5)+(c+a)(a−b)(6—c)+(a+b)(6—c)(

ca) (ab)(a—c)(ad)+(6a)(6c)(6d)+(c-a)(cb)(cd)

+(d-a)(db)(dc).

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5.

+

a

62

ca

6.

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XX.
Find the simplest forms of the following expressions :

b
b+c-a c+a-6*atb-0%
1
1

1 2.

1 6

+

6 3.

(c+a-b)(a + bc)(a+b-c)(6+c-a)+(6+c-a)(c+a-6) (a+c—b)(a +bc), (a+b—c)(6+c-a), (b+c—a)(a +6–6).

+ (a-ba-c) (6 -c)(-a) (c-a)(c-b) a(b+c), 6(c+a) c(a+b) 5. 6+crateta-+ +

a+b-o'
atb-ca-b+c
6.

(6c)
a-6+c a+b-ca-(6-c)*

Sa+bb2+-a? U Sa+b+c a+c-61 7.

atc ab

bc a?(6c), 62((-a), 6-(—b)? 8.

(c + a): –32+(a+b)2-6 a(6—c)? 62—(c-a) -(ab) 9. (6+2)—a* Teta c?

(a+b)sb, ,

+c-2a cta-2b a+b-2c 5-0
10.
be

+4-6): 16+c-2a+c+a-2c
ab

cta-
a-b
+

a+b^2c

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a-C

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XXI. Verify the equivalent expressions6

6 1.

+
ath a-6a-6 a+6
a +6

a -6° 4a2.

4: 57:+2+-7%=a

a (b:--)-ab(26+bc-(*) +*(b+c) alb-0)--) 3.

a (6+6)-ab(26° +3bc+c)+63(b+c) – a(6+c) -63

(a+b+d)(a+b)-c(c+l) a+b- -c 4.

(a+b+d)(a+d)-cb+c) a-c+d

da-6)(6-0)+ba-d)(c-d) 6-d 5.

ca-ba-d)+a(-c)(c-d)

(a-6)(5-0)+(b-c)(c-a)+(d-a)(a-c) 6.

(a—c)(cd)+(6d)(d-a)+(d-6)(6c) 6-d*

(a +y:)(6+xz)-(ab+cz) 7. (a +yz)(-xy)-(ac-by)" y

(a+bc)(b+ca)(c+ab)+(abc)(b-ca)(cab) 8. (1+a+b+c)"+(1+a-b-c)?+(1-a+b-c)+(1-2-6+cja

abc

2 (2b-c-a)-(20-a-6) 9(6-0) 9.

(c-a)-(a-6 6+c-2a 10. a'(b—c*)+b%(co—a)+c*(a*—12)_bo+catab

a'(b-c)+b(c-a) +ca-b) a+b+c

(caB*)(abc*)+(abc*)(bca~)+(bc-a)(ca62) 11.

(bc-a*)'+(ca64)' +(ab-co)

a +39 ++ ab +ac+bc

ab+ac+be (ab+cd)(a +b-c-d)-(ac+bd)(a −6+6—d) 12. (a’ +62—-—-)(ab?+c-do)+4(ab+cd)(ac+bd)

(a-d)(b-c)

(a+d)(6c)? {(a–1)x—(6—1)y}(axby)(ab)Pry 13.

ala-1):-(6-1) {(a-1)x-(6-1)y}(ay-br)-(a-b)ory ba-1).+a(6-1) 14. (2x*—-)+(2y——~)*+(23—Xy).

(y-2)+(2-x)+(2-y)"
(x+y+z)'+(3-y)+(x-2) +(y--x)?

x+y+
be('_6)+cala:-) + b(19-d)
15.
be(c-b)+ca a-c)+ab(6-a)

(a—62) +(6<->) +(6.4—a?):

(a−b)+(-c)+(c-a)

16. (x—a)+(7—6)+x_{s+m(x—a)+n(y-)}

1+mo+na (xmz—a)+(7—12—6)’+{n(x—a)—m(yb)}”.

1+m?tn?

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9

{, 2 }

}

(x+y)(1-xy) x(1-yo)+y(1-x*) 3.

(1-xy)–(x+y)(1 —x*)(1-y?) — 4xy 4. 1+

1-x , (1-x)(1-x*), (1-x)(1-xo)(1—29)

1+x+(1+x)(1+x+)+(1+x)(1+2+)(1+x1). 5. { ax'-ay' +2bxy 1 by:—bxo + 2axy

+

x+y 6. (aybx)8—(ax-by)s , (aybx)+(ax-by),

(a+b(y-2) (a-6)(y+x) x?+xy+yo S1,1 , y'+ya+z. si, 11 7. xty

{+1}
x+y2
x+xz+1 1
ma

+
2x .2" 2y

1 1 I 3 1 2 8.

+ ;

+-+
2+y x-y

y
Y

-yy
Sa+bx btax a+bx b + ax
9.

+

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.

C

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{

+

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+

ab

bc

ac 4x?

{ab-1-(a - b)x}" 10. ab-1-(ab)x}-{a-6-(ab- 1)x}?

{ab-1-(ab)x}? *(1-ab){1+ab+(ab)x}-(a−b){a+6+(ab+1)x}" 11. (a+b)(a +6:—<^) (6+C)(b +6* – a") (a+c)(a*+co – b).

+ 1 1+x

1+"

1-2
1**+173}={1**+147-173}
U(y+z)

(+)
}
1 (x - y)

}:{
(y-2)

(8 x
*{-1}{*}{-1}.
14.

(x+2y)ox (2x+y*y
{_
(2x+974

{ leto 2015* +420 7 }
(3-y) -y)

(x+y)'

12.

+

5 (x+y)+1

13.

+1

*3)*+1}

Х

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