26.

б.х? - 17ax + 12а?
4.х 50
24х3 - 68ах2 + 48а?х

- 30ах? + 85a3х – боа? 3* - 4a)247 – 98ax: + 133a** - 6009 (8.12 – 222x + 1522

боа" (8424х3 - ?

28. x2 + 4ax + 3a?).*+ 30x' + a*x + 3a*(*2 – ax + a?

за** -
2х + 4ax + 3°х?

ах – 3g?х? + a*х
ах - 4а?х? - зах

а?х? + 4a3x + за
а?х?+ 4a3x + 3

If as

2. x2 + (x2 - 42x+89)}, when x = 2,

=4+(4 - 84+89)* = 4 + Vg24+3=7 or 1.

at sa2 +62 a3 - 26(a? - 62)

- b, and b= -3, -b+ 202 +22 -bt/262 - 13 - 26(62 - 62) -63 6- 262 63

=(3+ 118) - 27

= (1 + 12).

3. (a) By x – 50.
(6) * number of hours.

у
(C) x2 + bax - 91a2=(x - 7a) (x + 13a).

4. (2 + 3x + 4x4)2-(2 - 3x + 4.x2) = (4 + 12x + 25x2 + 24.x3 + 16x4)

- (4 – 12x + 25x2 – 24x3 + 16x4) = 4 + 12x+ 25x2 + 24x3 + 16x4 – 4 + 12x – 25x2 + 24.78 – 16x4 = 24x + 48.x3 = 24*(1 + 2x2).

a

5. The parts of a compound expression connected by the signs + and - are called terms-e.x, in a2 +62 -, a, b, and

6’ ( are terms.

Coefficient=co-factor. ac is produced by a xc, in which each of the factors a and c is a coefficient of the other.

An index is a number placed over a quantity to show how many times it occurs as a factor in a power; thus, in 33, 3 is the index which shows that x occurs as a factor three times, as X X X X X.

A negative quantity is a quantity with the minus sign prefixed, as - a.

In the quantity abc, a, b, c are factors, because they make by multiplication the quantity abc.

12x2 + xy - 63.72 = (3x + 7y) (4x - wy).

6. See No. 5.
as +63 + 3(a+b)ba = a + zaob +3ab2 +68

= (a + b)

7. (12 + xy + y2)2 = (x2 - xy + y2)2 + 4xy(x2 + y2),

+ ,
or *4 + 2xy + 3x2y2 + 2xy + yt
= 44 - 2x3y + 3x2y2 - 2xy + y + 4.xoy + 4.248

x4

= x+ + 2x3y + 3x+y2 + 2xy + y^. If x = 5, and y= -5, show that

(x2 + xy + y2)2 = (^2 - xy + y2)2 + 4xy(x2 + 3,2).

Substituting the above values,
(25 - 25+25)2 = (25+25+25)2 - 100(25+25),

or 252= 752 - 5000,
625 = 5625 – 5000,

= 625.

8. Show that x(x + 1) (x + 2) (x + 3) + 1 = (x2 + 3x + 1)2. Multiplying out, and squaring opposite side, we have

*4 + 6x3 + 11x2 + 6x +1 = x4 + 6x3 + 11x2 + 6x + 1.

9. Let a? – b2 be divided by a +b, where a+b is the sum of the two quantities a and b, and a? - 62 is the difference of

their squares.

Here the quotient is a - b, which is the difference of the two quantities a and b. (aox? + b2y2) - (a2b2 + x2y2)= x(a? - y2) – bo(a? - y2)

ax + by + ab + xy = x(a +y)+ b(a + y). 3b *(a + y) + B(a +y))x?(a– y2) – bo(a? - y2)(x(a – y) – b(a - y) Ix®(a2 - y2) + bx(a2 - y2) = (x -b) (a - y)

3–
-- bx(a? - y2) - 64(a? - y2)
- bx(a2 - y2) - ba? - y2)

10. a +a+a+etc. 6 times = ba.
Let 3 and 4 be two consecutive integers,

then (3 + 4)2 – 1 = 3 X 4 X 4 = 48,

72-I = 48 49 - 1 = 48.

=

11. X-(a - b)x3 + (a - b)028 – b4; find value when a= = , and b=0.

As b=0, all the terms which contain it vanish, and the expression becomes

#4 - ax: or x®(x - a), which, as a = - *
=**(x+x), or 2x*; and - x= 1, x= -1;

.. **= 1, 2X* = 5
24

.

1

SIMPLIFICATION. 1. ala – 1) (a – 2) (a - 3) = 24 - 6a3 + 1122 - 6a. If a = 4, the expression becomes 4.3.2= 24.

2. 4.x -[(4x - 4y) (4x + 4y) -- {4x +(4x + 4y) (4.x – 4y)} + 4y]

= 4x - [16x2 - 16y2 - 4x + 16x2 – 16y2} + 4y]
= 4x -[16x2 – 16y2 – 43 – 16x2 + 16y2 + 4y]
= 4X – 16x2 + 16y2 + 4x + 16x2 – 16y2 – 4y.
Collecting = 8x - 4y. If x= 1, y=0,

-8.

I

3. x(x + y) +*- {x(x - y) + (x + y) (x - y) - 66}

***+ xy + x - x2 - xy + x2 - y2 - 66}
= x2 + xy + x - ** + xy - x2 + y2 +66
= 2xy + x - ** + y2 +66, which if x = 100, y = 50,
*** 10,000 + 100 - 10,000 + 2500+ 66
= 2666.

4. See Mansford, page 13.

3(x2 + y2) - {x2 + 2xy + y2 - (2 + y - 42 - y2))} = 3x2 + 3y2 - {x2 + 2xy + y2 – 2- y + x2 + y2}

3.x2 + 3y2 - ** - 2xy = y* + 2+ y - x - y Collecting = x2 - 2xy + y2 + y + 2.

5. (a + b + c)2 - a(b+c-a) - bla+c-) - ca+b-)
=a+ 2ab + zac + b2 + 2bc + ca - ab - ac ta? - ab-bc +62

- ac - bc +62
= 2a +262 + 20. If a-1, 6-2, 6-3,
* 2+8+18= 28.