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15.

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a- 6 a-6

(x+2y) (2x+yy!
16.
(x+y)' (x+y) (-y) (-y)

XXIII.
1. If x= a +b+c+d;
8-a
8-6

s1 1 1 1
then
-+

8-4.

+
6

C d
2. If s=a+b+c+d+ to n terms,

-6

8-C then shall

+
+

=N-1.
8
3. If xyz = a', y-zx=b?, z'—xy = co,
find the value of

a'r+by+oʻz

in terms of a, b, c.
x+y+z
6-0 C-a

- 6
4. If x=

; then xyz+x+y+z=0.

b
5. If y+:+u=ax, :+u+x=by, utx+y=cz, x+y+s=du;

1 1 1 1 then shall

= 1.

cti++i
6. If a =

b = Y-3
+y y+z +3
1ta 1+1+0

1.
1-a1-6°1-0
6—c0a)

a

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prove that

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8. If 28 = a +b+c,
1 1 1 1

abc then shall

+ +

+

8-0788(8-a)(8-6)(8-6)*
12+(-a' ta- a + b --
9. If

+
2bc

=l;
+
ас

2ab then shall (a+b-c)(a +(-6)(6+0-a) = 0.

a + bc 1 +ca +ab 10. If

= 1, a, b, o being each greater than 0;

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4. If x=

and y=

and y

4

+

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a

с

a

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and ascertain if the value of the difference is dependent on the value of x.

2. If xyz=1, then (1+x+y)+(1+2+x-1)-+(1+y+z=')-'=1.

3. If a =(b+c)x, b =(c+aly, c=(a+b); then 1-xy-yz-23-2xyz = 0.

a+
a+b-c a+bx (a-b+c)' +4ab

then
atoto b+cx? (6 - a+c)' +4ab'
2ab +62

a? - 72 5. If x=

;

then 2 + y = y$+$. a'+ab+

atab +62 6. If x= a +6+ (a-b)

a+b

ab 4(a+b)

tato: shew that (x – a)-(4-6) = b». 7. If x+y+z = 0; then *(78—z), y(3°—29), (2— y")

+

-0. Y8. If 672-d+y

shew that either number is equal to

d be ad dx by 9. Of the fractions (a+b)(c+d) (a+c)(6+d) (a + 2)(b+c)

ab+cd

ac+bd ad + bc if any two be equal, the third is equal to either of them, and each to-1: a, b, c, being unequal quantities. 10. If (ab+ac-bc)(bc + acab) (ab + acbe)(ab+bc-ac) ab+bc-ac

bctac-ab (ab+bc-ac)(bc+ac-ab)

; then shall a=b=0. abtac-bc 11. Shew that for all values of x, the sum of these two expressions

(x-a) (1-6) (x-c) (ab)(ac)+(6—c)(6a)+

and

(c-a)(-6) 1 1 a -6

+

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+

ame} (x-ast

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хс

}

1 +

(x-c), is equal to a constant negative quantity;

C-6 supposing a, b, c constant and x variable. 12. Determine the value of the fraction

r-2.ro +3

when x=

= -1.

2013 + 3x2—2 13. What value of x will cause both the numerator and the denominator of the fraction 12.08 — 23xoa +19xa’-6a8

to vanish, and what is 9.03 — 21x+a+22xa? - 8a the true value of the fraction in that case ?

rs + 6x + 13x + 10 14. Reduce

to the form of a continued fracx+63

+14r +152 +7 tion, and verify the truth of the result by reversing the process.

1ta ?

or

a

6+

a

less than a+nb

a

4. If

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bers; shew that ma+ne

Or

ay+bx

1

XXV. 1. Which is the greater fraction,

1 tao
1-a

1-a
b

1 1 2. Shew that

+ is always greater than

62 a? 3. If m be less than n, then shall a+mb

be greater than, equal to, or

b+ma
, according as a is greater than, equal to, or less than 6.
b+na'

and be any two fractions, and m, n be any integral num-
6 d

nc

is intermediate in value between , and mb+nd

a 5. Which is the greater fraction ar-by

aybx?
ar+by

26 6. Prove that is always less than

if b be intera+c

(a+b)(b+c)' mediate in value to a and c.

7. What are the integral values of x, if (x+2)+ be less than +4)

:+4)+3, but greater than (x+1)+22
8. If x+? be greater than 3 : then shall x+?> 3.
9. Shew that the sum of every fraction and its reciprocal is equal

bd
to, or greater than 2, and that ;+:+ă+ätä+ >6.

10. If a, b, c be unequal, then a'(b+c)+b+(a+c)+c+a+6)> 6abc.
11. If a, b, c be three quantities in order of magnitude,

a 6 b
then shall + tī>tit

6 a
1

1

9 12. Shew that b+c-a

+

cta-b7a+b-c a+b+c 13. If a, b, c be such that the sum of any two of them is greater

b than the third ; shew that

+

> 3. atb

a

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a

c

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a-6

a

6. If + = 1 and

с

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a+bz

e+ da provided that

a

a-d

4. If arta' =bx+b' = cx+c'; then ab' ta'c+bc' = a'b + bc+ca. o-d

b-C

d-a 5. If

0, prove
that

0.
itab+ited

1+be+itad

XY
then
1,

-1.
+
6

abc

1etz

1-e 17. If x =

then shall 1+ez'

1+x1 te 1+: at bx 8. If y=

a+by and %=

; then shall x = et da

c+dy b++ad+bc = 0.

1 1 1 1 9. If aʻ(b—_)_b}{a^), then

b-d

+i= +a; and conversely.

a 73 10. If (a + b)x? = 1, then shall + (a+b)(a +b).

1-bx1 tax 11. If xy = ab(a+b) and x?—xy +y =a +63, shew that

1 x
Y

y
b 6

12. If

x=-
-y2 xy

ys, then shall
B

y ad-bc

- bd 13. If

then atb=c+d, and each fraction a-b-c+da-6+c-d' is equal to 1(a+b+c+d). 1

2 x 1 x, y 13

yl 14. If

then shall each +

+ +

1 % Y

s Y

XY 15. If

y +

0, then x+y+z=0. 6 b

1 1 1
16. If

0; show that x =y=.
+ +
Y

y
17. If

then shall ax+by+c=0.
b

1
1

1

0. 18. If ab+ac+bc = 0, then

+
a'-bc

it
62

-ab

-0.

a

a

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%

I 19. If

a,

= b, =C,
yt:
x+%

3+ y

22 then shall

a(1-bc) (1-ac) c1-ab) x+y- y+ - x

x+:-Y
20. If

then a = b = c.
ax+by-czbytez-acax+cz-by'
if
+ +

= a,

y and

-- b, then ab=9, if x+y+* = 0. stx'xty

21. Shew that if {*#*++*+*+/

} }

y
to +

%

y t

RESULTS, HINTS, ETC., FOR THE EXERCISES ON COMMON

MEASURES AND COMMON MULTIPLES.

I. 1. See Section V., Art. 2, pp. 1, 2. 8. The numbers 12 and 6 ; 4 and 18 ; 2 and 36, respectively fulfil the conditions. 9. 34 + 12x3 + 49x? +82x + 48.

II. The following are the highest common divisors :1. 3x +1. 2. 2-1. 3. X-1. 4. 22-y. 5. &-a. 6. 2? + 2x. 7. 3x2 -- 2. 8. 33 + 3.02 + 3x+1. 9. 2-1. 10. 2+2. 11. 2? - 1. 12. 2-a. 13. ay+ab. 14. - 1.

15. 1-ax.

16. x+a+b. 17. a+b+c. 18. (x - 1)? NOTE.—In the examples numbered 2, 3, 5, 9, 11, 14, 15, 18, the highest common divisors may be found by inspection. See Section IV., Art. 13, p. 24.

III. The following are the least common multiples : 1. n(n − 1)(n − 2)(n - 4)an-1x4. 2. 12.04 – 412S + 25x2 +16x – 12. 3. a(x+a)(x - a)?(x2 +a”). 4. 3x6 - 1125 -- 8x4 + 4223 +19x2 – 47x - 30. 5. (204 + 3a36+3a2/2 + 3ahs +264)(as + 3a4b+ab? ta’83 + 3ab4 +65). 6. (a2 + ax +x2)(a? – xo). 7. ** - 16a". 8. (3x - 1)(4x - 1)(7x - 2). 9. (2:2 - y2)(x+ + x2y2 +y). 10. x4 +10x3 + 35x2 +503 +24. 11. 36(a? +62)(a3 – 63)(a - b)3. (a+b). 12. x(x + 1)(x2 + 1)(x2 - x + 1). 13. x(x+1)(x2+x+1)(x+ - 1). 14. (2x - 3)(3x − 2)(4x + 1)(5x +4). 15. See Exercises XIX., No. 4, p. 35.

IV. 1. a=8, the highest common divisor is x2 - 4x +3, and the fraction in its lowest

X-4 terms is

2 -5 2. It may be shewn that x3 +4+z3 = - 3.xyz, and 25 + ys +75 = 7xyz(yx —x?). The truth of this may be verified if x =2, y=3, z= -5.

3. At the third step of the process for finding the highest common divisor of the two quantities, it appears that the question is reduced to ascertaining if an-2 - bm-2 is divisible by a2 + ab + b2.

4. It is sufficient to remark that the quotient of an odd number divided by an odd number is an odd number, and the sum of two odd numbers is an even number.

5. It is possible, that when numerical values are assigned to the symbols of two algebraical expressions, and the greatest common measure of these numbers be found, the result may not be the same as that found by assigning the same numerical values to the symbols of the highest common divisor. This may arise either from the relation of the numbers, or from the introduction or removal of factors in the process of finding the highest common divisors.

In the first example, the highest common divisor is x2 +x+1. When 4 is substituted for in the two given expressions, their values become 231 and 399, and the greatest common measure of these two numbers is 21. And when 4 is substituted for x in xo +x+1, the highest common divisor, the result is 21, which is the same as the greatest common measure of the two numbers found by the same substitution.

In the second example, the highest common divisor is 3x+ 4a. When 4 is sub. stituted for x and 1 for a, the two expressions become 48 and 192, and their greatest

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