7. (a3+b3)(a3—a3b3+b3). 8. (a3—b3)(as+a3b3+b3).

9. (a2+a3b3+b3)(a3 — a3b}+b3).

10. (x*—2x+3)(x*+2x+3).

Determine the following quotients, and verify the truth of them :1. a-b by a+b*. 2. a3-b3 by a1—ba. 3. a2+b2 by a+2*a*b1+b. 4. a1+b1 by a}+b§. 5. al—b§ by a1+b1. 6. 3ao—4a{b}+b§ by (al—bi)2. 7. xl−31x+9x1—3} by x—3. 8. x+yl-l-2xyl by x-y}+zi.

9. x-x2y+xly—xy +x1y2—y} by x2+xy+y2.

10. a2ab+b2 by a-31ab+b. 11. x-4x+12x-9 by x-2x+3. 12. x1+1 by x2—21x+1. 13. 2x3-6x+5 by 2x+23+1.

14. x'—y' by x1—y1.

Divide

-y by x-y, and x-y by x-yt, and from the results infer the quotient of a-y divided by x-ył.

XX.

Verify the following equivalent expressions:

1. (x—21x1+1)(x+21x1+1) = x2+1.

2. (a2+21ax+x2) (a2 − 21ax+x2) = a*+x1.

3. (x2+3*x+1)(x2—31x+1)(x−21x1+1)(x+21x1+1)= 2o + 1. 4. (x2+21x+1)(x2 −2*x+1)(x2—21x—1)(x2+21x—1)

=x-4x+2x-4x2+1.

5. (x-1){x+(1−√3)}.{x+(1— √√/3)} = x2+x2-4x+2.
6. {x2+1(1+√5)x+1}.{x2+1(1-5)+1}=x+3x2+1.
7. (x+√2√3) (x−√2−√3)(x+√2+√3)(x−√2+√3)

9.

(2a*c*+b2)x2+2a}bx}+ax3+2bc{x}+c3x.

10. 1+x to four terms. 11. 1+x+x2 to five terms

12. a2+ab+b2 to four terms.

XXIII.

Verify the truth of the following identical expressions:—

1. (x§—1+x¬3)(x}+1+x ̃3)=x3+x¬3+1.

2. (2x+y-1)(2y+x−1) = (2x1y*+x ̄1y−1)2.

3. (x*—3y1)(x*+y*)(x*+2y*)(x+3y1)(x*—y1)(x*—2y1)

=2-14x3y+49xy-36y'.

4. {(1+x)(1−x)*—(1−x)(1+x)*}*—{(1—x)*—1}2

= x2 { 2( 1 − x2)3 — 1}.

5. {xy+2x(xy—x2)*}1+{xy—2x(xy—x2)1 } = (2x2—xy)'.
6. {2x2+2(x1—y1)*}*—{2x2—2(x*—y*)*}* = 2(x2—y°)*.
7. (a1+b*—c1)(a1+b1+c$)+(a1—b3 — c¥) (a* — b1+c1) = 2(a+b+c).
8. {(ab)*+(ac)'+(bd)*+(cd)*}.{(ab)* — (ac)* — (bd)*+ (cd)*}

9. (a®+a1b2)+(b® + b1a2)1 = (a2 + b2)§.

10. (a*b+a3bc)*+(ab*+b*c)* = (a+b)(a+c)*b*.

11. (a2+a3b3)+(b2+bša3)* = (a}+b1).

= (a−d)(b—c).

12. {(a+b)}+(a—b)3}(a+b)ŝ—{(a+b)$—(a—b)$}(a—b)3 = 2(a2 + b2).

XXIV.

Find the values of the following expressions:

1. {(x2+y3)*+z}.{(x2+y3)i—z}, when x=4, y = 5, x=6.

24

2. (1−x)*+{1−x+(1+x)'}', when x= 25

3. x3-2x2+x+18, when x=2+√−5.

4. {x1+2x3+3x2+2x+5}', when x2+x+1=0.

5. x+y+(x2—xy+y')', when x = √2+1 and y=√/2—1.

when a = - · 4 and b=

6.

a+(a2+b2)+
a3 — 2b(a3 — b2)3

x-y

-3.

8. *+*+y, when x=2+√3 and y = 2-√3.

x+yx-y'

g.

(x2+a3)*+ (x2 — a2)*

x2+xy+y, when x=
√3+1
√3-1
x2-xy+y2'

10.

(x2+a3)‡ — (x2 — a2)i

Reduce the following expressions to their simplest forms:

1.

2.

x+6x2+5x3

XXVI.

Verify the correctness of the following equivalents:1. xy { x2+ y2+ y2+x2

x2+y2,

y2

= x2+ y2.

_

' } * = x*y*(x2+y3)• ̧

y

27a-18ab2 — b4
8ab2

x+y
abs

bc

8ab2

a2

6. (x+y)x* _ (x−y)y* _ (x+y)y* _ (x—y)x* __ x3

1

(1+x)}' (1+x)}' (1+x)} (1+x)

(1+x)

8.

√2-1 +1

2√2 x2+2*x+1

9. (1+√5)x+2

x2−} (1+√5)x+1' x2−1(1−√5)x+1

+

(1-5)x-2

2+1 23+2x2+(1+25)x x1—x3+x2¬x+1

10.

x+1 X-
X

X

sx-1

x+1

x+1

=

b+x (b+x)} (a2—x2)}**

15.

2-2.x2

1

16.

=

{(a—b)2+4ab}'.{(a+b)2—4ab}?. { { a = b2 - 3ab } }

(1—x2) (1—x2) (1—x2)* *

17. (1−x2)*(1+x)2 { (1−x)1 ̧ (1+x)1 )

a

.

= a2-b2.

(1+xji+(1+x); } = (1−x2)(1+x)?.

'

XXVII.

1. For what reason is the negative sign necessary for distinguishing algebraical from arithmetical calculation? and how does the algebraical representation of impossible quantities result necessarily from this use of the negative sign?

2. Is it always possible to express the sum of the squares of two algebraical expressions in the form of the product of a sum and difference?

3. Shew that in the addition, subtraction, multiplication, anl division of quantities of the form a+b√-1, and also in the involution and evolution of such quantities, the results will always be of the form AB-1.

Express

and

2+3/-1 a+b√-1
4+5-1 c+d√-1

-

in this form.

4. Explain why the introduction of imaginary expressions does not vitiate a process of algebraical reasoning.

5. Shew that {-1}" will have four different values according to the forms of the number m, and exhibit the values of (1+-3).

6. Shew that (-a)TM× (—b)TM is always an impossible expression,

and (—a)îm'+î× (—b)TM+ is always a possible expression, when m is any integral number.

7. Are the quantities /-a × /—b, and /—a× —b, rational or irrational?

8. Shew the absurdity of the following reasoning:

(−1) = (−1) = {(−1)2}' = (+1)' = +1.

9. Point out the fallacy in the statement

√(−1) × √(−1) = √ {(−1) × (− 1)} = √ +1 = 1.

10. Find the third power of

1-3, and of +1-3; what

inferences may be deduced from the results of the operation? Shew

11. Shew that any power of a cube root of unity, is itself a cube root of unity. Exemplify this property in the fourth and fifth powers. 12. Is the following argument legitimate? If ao1 and 6o = 1; therefore ao 6o, and consequently a = b.

=

=

13. Establish the equalities, in which is an imaginary cube root. of unity,

(y-x) (y+-2x)+(≈—x)2(z+x-2y)+(x-y)(x+y—22)

= −(y+z−2x)(z+x−2y)(x+y−2z)

= (x+wy+w2%)3+(x+w3y+wz)3.

XXVIII.

Verify the correctness of the following expressions:

1. √(—a3).√(—b2) = —ab,

and √(—a2). √ (—b2). √√ (—c2) = —abc/—1.

2. (―a3). (—b3) =+ab, and /(—a3). ✅/ ( — b3). 3/ ( —c3) = — abc. 3. {−√(−2√ − 3)}' = −12.

4. (a+b-1)(a−b−1) = (a+21a1b1+b)(a−21a*b*+b).

5. (a+b2-1) (a2—b3/−1) = (a2+2ab+b2)(a2—21ab+b2). 6. (a+b-1)(c+d−1)(c—d−1)(a−b−1)

=

· (ac+bd)2+(ad—be)3.

7. (x−√2−√−3)(x−√2+√−3)(x+√2−√ −3)

(x+√2+√−3) = x1+2xˆ+25.

8. (1+1)+(1--1)=0 and 4-1.

9.

10.

(16+30-1)+(16-30/-1)=10.
{a+b-1}+√{a−b√−1}= √{2a+2√(a2+b2)}.

11. (a+b√−1)3±(a−b √ − 1 )3 = 2a(a2—3b2), and 2b(3a2—b2) √✓ — 1.

a3. -(1−√ − 1)a2b+(1−√/ −1)ab2+b3√ −1 = a2 — ab+b2.

a+b√-1

a-by-1

2.

3.

=

1+2-1 1+2-1

4.

8√-1

5.

a-b-1, a+b-1 2(a2-b2)

a+b√-1a-b√1 ̃a2+b2

1+2/-1 1-2-15

1+(1-√2)√-1, 1-(1-2)√−1.
1 − ( ' — √2) √ −1+

=

1+(1+√2) √-1

√2.