17. 3.38 – 10x® +15x+8) ** – 234 – 6.3°+ 4x2+13++

+ 6

3 3x5 – 6x4 – 18x3 + 12x2 +39x + 18/1 - 10x3

+15x + 81 - 2) - 6x4

8x3 + 1 2x2 + 24x + 10 3.34 + 4x3 6x? – 12x

5

3.75

3x4 + 463 – 6x2 – 12x –

+5)3-7) – 10x3+156 + 8

+ (*
- 4) - 4x4 - 4x3 + 12x2 + 20x + 8
*4 + x3

3 - 3x2 – 5*

- 2

I 2X

*4 + 33 – 3x2 – 5x – 2

5/3 3x4 + 3x3 – 9x2 – 15x – 6

X3 + 3x2 + 3x + 1

2)33* + 4x4 – 6x?

*3 + 3.x2 + 3x + 1
+ 1** + 33 – 342 – 5x – 2(x + I

(1
1x4 + 3x3 + 3x2 + x
-2) – 2x3 – 6x2 - 6x – 2

H3 + 3x2 + 3x + 1
x3 + 3x2 + 3x + 1

.. G. C. M. = x3 + 3x2 + 3x + 1.

18. Let c be contained p times in a, and 9 times in b, then a =pc, b=9c, and ma nb=mpe + nqc=(mp + ng)c; hence c is contained (mp+ng) times in ma + nb, and therefore c measures

ma + nb.

N.B.—Some error in the copying of this example or in the setting of the original.

20. For proof of this rule see any text-book on algebra.

Here G. C. M. of 21 and 28= 7,
and x2 - xy + y2 is a measure of x6 – 36 ;

.: G. C. M. = 7(42 - xy + y).

21. (aob - ab2)2 = abo(a - b) = ab(a2 – 62)2

=abla? – 62) (a? – 62) = ab(a - b)2 (a + b)2;

.. G. C. M. = ab(a - b)?,

6(x2 - 1)=6(x - 1) (x+1),
8(x2 – 3x + 2)=8(x - 1) (x - 2);

.. G. C. M=2(x - 1).

22. 4(1-3 +93) = 4(x + a) (x2 – xa +a),

6(x2 - 2ax - 30%) = 6(x + a) (x - 3a);

.. G. C. M. = 2(x + a).

LEAST COMMON MULTIPLE.

1.

(x2 - a2)=(x + a) (x -a),

(x3 - a*)=(x-a) (x2 + ax + a); :. L. C. M. = (x – a) (x+a) (x2 + ax + a2) = x4 + ax3 – mox – a4.

2.

(x+2)=(x + 2),

(x2 - 1) = (x - 1) (x+1),
(x2 + x - 2)=(x - 1) (x + 2);
i L. C. M. = (x + 2) (x - 1) (x + 1)

= x3 + 2x2 – X – 2.

3.

15a2 + 16ab - 1562 = (3a +56) (5a - 36),

9a2 – 25b2 = (3a + 50) (3a – 50);
.: L. C. M. = (3a + 50) (3a - 56) (5a - 36)

= (ga? – 2562) (5a – 36).

4.

+3 + x2 + x + 1 = (x + 1) (x2+1),
x3 – x2 + x – 1=(x - 1) (x2 +1);

L. C. M. = (x2 - 1) (x2 + 1)=(x4 – 1).

5.

12x3 + 4x2 – 3x – 1= (3.x + 1) (4x2 - 1),
8x3 - 4x2 – 2x + 1 = (2x – 1) (4.42 – 1);

. . L. C. M. = (3x + 1) (2x – 1) (4x2 – 1).

6.

x+ + xy2 + y4 = (x2 + xy + y2) (x2 - xy + y2),

33 - 33 = (x - y) (x2 + xy + y2); .: L. C. M. = (x+ + x2y2 + y^) (x - y).

7. Let a and b represent the two algebraical quantities, and d their G. C. M., and let a=pd, b= qd, so that p and q have no common factor. Then the least quantity that contains p and 9 will be pq, and therefore the least quantity which contains pd and qd will be pod, which is therefore the L. C. M. required of a and b.

2.

** + xy **(x2 + y)
*4 - y2 (x2 + y) (x2 - y)** - y'

x2 + 3x + 2 _ (x + 1) (x+2)_x+I
x +3 - 2 (x - 1) (x + 2) * -I

3.

m

1 1712 + 2 2mx Im(m + 2x)
33(m2 – 432) 33(m – 2x) (m + 2x)3(m – 2x)
a+ 3a2 - 4 _ (a – 1) (a? +42+4) _a2 + 4a +4
as

(a – 1) (a” +a+I) ao tati

4.

a + (a + b)ax + bx2 Q3 + a'x + abx + bx?
al - 62x? (a? – bx) (a + bx)

(a + x) (a2 + bx) atx
(a? – bx) (a? + bx)a? – bxo

2a

Simplification. 1. }{x(x + 1) (x + 2) + x(x - 1) (x - 2)} + 3(x - 1) a(x+1) = }{x(x2 + 3x + 2) +x(x2 – 3x+2)} + (x - 1)

)
= }{*3 + 3x2 + 2x + x3 – 3x2 + 2x} + gax? – şa
={{2x3 + 4x} + gax- ja
3x3 + 3x + žax? – fa=}{x3 + 2x + 2ax? – 20}.

3

=

2r2

al - x2-1
a- ax + ax + x2 – 2x2 - a2 + x?

5.

5.5
a x(a2 – 62), ax?(a2 – 62)

+
62 62(6 + ax)
ab(b + ax) - x(a? - 62) (6 + ax)+ axa- 62)

62(6 + ax)
ab2 + a2bx – a3x2 – a+bx + abx? + 13x + a’x? – ab? x2

bo(6 + ax)
ab2 + 33x a + bx
64(6+

ax) 6+ ax

a3 – 63

+22
-as + 38 a2 + ab +62

b(a? +62)
a+b +63 a2 + ab + b2 ba? +62) a2 + ab + b2
a(a3 - 63) ba? + 62)a(a3 – 68) bla? +62)

a2

{09+a=

a(a - b)