Lessons in Algebra.

(For Pupil Teachers in their Fourth Year.)

BY THE EDITOR.

CHAPTER VI.-Theorems derived from Division.

30. The student should carefully notice and verify the following Examples:

In the first pair of examples, the index is an odd number (5); in No. 1, the terms of both dividend and divisor are separated by a minus sign, in No. 2 by a plus sign; the two answers or quotients are identical, except in signs;-in the first case, in which the divisor contains a minus sign, all the signs in the quotient are plus; in the second case, in which the divisor contains a plus sign, the signs in the quotient are alternately plus and minus. A little attention will show, too, that in all the Examples the powers of the first term decrease regularly in the answer; while the powers of the second term increase in exactly the same propor tion. We may condense the foregoing statements as follows:

When the index (n) is an odd number, a+b" (read a to the nth, plus b to the nth), is divisible by a+b; and a" - b" by a -b. When the index (n) is an even number, a" - b" is divisible by both a + b and a b.

When the divisor contains a minus sign, all the signs in the an swer are plus; when the divisor contains a plus sign, the signs of the answer are alternately plus and minus.

The first term of the answer is obtained by dividing the first

term of the dividend by the first term of the divisor; the last term of the answer may also be obtained by dividing the last term of the dividend by the last term of the divisor.

Before proceeding to the following Examples, the student should carefully re-read the preceding statements until he is thoroughly familiar with them.

31. Each of the terms may be composed of several letters instead of one merely, as in the following:

Ex. 6.

{

2x

16x4 - y+24

yz

a4b4 — a3b3. c + a2b2. c2 − ab . c3 + c4 .

= 8x3+4x2yz + 2xy2x2 + y3z3. In this

example the 16 in the first term of the dividend was the 4th power of 2, and might have been written 24; the whole Example, indeed, might have been written thus:

In the following a fraction is used in the same way :

4. x5y5+z5 by xy+z; and x3у5 - 25 by xy-z.

5. a1-b4c4 by a + bc, and by a − bc ;

6. m3ï3 +n3y3 by mx+ny; and m3ï3 – n3y3 by mx – ny .

7. 16a-b4c4 by 2a + bc, and by 2 a-bc.

8. a5-32c5d5 by a -2cd.

9. 81x4y-1 by 3xy-1, and by 3xy+1.

10. m3 27m3 by 1m+3n.

11. 8x3y3 3 by 2xy - 3z.

12. m1 - n1 by 3m+n, and by m – 1n.

32. The following additional Exercises may be worked by a careful application of the same principles; students to whom the subject is new may miss the remainder of the chapter, however, as the examples to be dealt with are somewhat difficult, and for most ordinary purposes are not necessary.

a4-16b8
a +262

Example 1.first term in the dividend is 4, that of the second term is 8; all that is essential in this respect is that the first term in the dividend should be the same power of the first term in the divisor as the second term in the dividend is of the second term in the divisor. For example, in the above the first term in the dividend is a4,-that is the fourth power of the a in the divisor; the second term of the dividend is 1668, that is the fourth power of the 262 which forms the second term in the divisor. The example a4 - (262)4 might have been written ; the answer thus: a + (262)

-In this example the index of the

a3 — a2(2b2) + a(262)2 – (262)3, which equals a3 - 2a2b2+4ab1 — 8bo.

Example 2.

ab+b6
a2 + b2

Here the indices are even numbers,

and the sign between the terms being plus, it looks as if the upper line could not be divided by the lower; but the a6 and 66 are respectively the third power of the corresponding terms in the divisor, and this index of course would be an odd number.

example might have been written thus:

(a2)3 + (b2)3

a2+b2

The

and the

answer thus: (a2)2 – a2b2 + (b2)2; which equals a1- a2b2 +b4.

Example 3.

-In this example the index of the y in

the dividend is 4, but this is in reality the third power of the corresponding term in the divisor. The example might have been 23 - (y)3

written:

; and the answer thus +x2+xy+(y‡)2, which

x - y3 equals x2 + xy+y§. .

a5+b5

Example 4.

Here although the indices in the

a+bi

dividend are odd numbers, the upper line is not exactly divisible

by the lower, for each term is the fourth power of the corresponding term in the divisor; the a3 is equal to (a)1, and the 65 to (b). The example might therefore have been written thus: (ak)4 + (b?)4.

a+b

20+ y

; or, if a be written for a, and y for b, it would be

20

; in either of these two forms it is easily seen to be a case x+y in which the upper line is not exactly divisible by the lower. The student should apply the theorems in paragraph 30 in the way we have done with the preceding examples.

The indices being literal instead of numeral need not cause any difficulty if the student add or subtract them carefully. 23+(a - b)3y3 Example 7.x + (a - b)y

= x2 — x. (a − b) y + (a − ¿)2 y2.

In this example the a-b is bound together by the bracket, and so regarded and worked as one whole; the brackets may, of course, be wholly or partially removed during the working, and the answer would then be x2 ·axy + bxy + (a2 — 2ab+b2)y2,— or x2 - axy+byx+a2y2 - 2aby2 +b2y2. The next is an example of the same kind.

4a3 + 2a3(b1+c)x+ + (b + c)2x,

8a - (b+c)3x

which equals 4a+2a3bx1 +

EXERCISE XII.

2a3cx + b2x2+2bcx2 + c2x2.

Write the quotients of

1. x5 - y10 by x − y2; and x1o + y5 by x2+y.

2. a3 – ba by a3 – b, and by a3+b.

3. m2 – n3 by m3 – n.

AUTHOR OF "LEADING EVENTS IN ENGLISH HISTORY."

PLANTAGENET PERIOD.-PART IV., 1272 тo 1327.

Edward 1st.-(Note 1) 1272-1307 (Longshanks).-Eldest son of Henry III.; in Palestine at time of father's death. Why? (Note 2).

Conquest of Wales, 1282.-Causes :-(1.) Ambitious policy of Edward, whose ruling idea was to unite the British Islands in one kingdom. (2.) Refusal of Llewellyn, Prince of Wales, to do homage to Edward.

Although ruled by a brave prince, the Welsh themselves disunited. Two of the brothers of Llewellyn, David and Roderic, ent all the weight of their influence to Edward.

English advanced to Snowdon-last refuge of Welsh army. Llewellyn obliged to surrender, and submit to terms of victor, which virtually meant subjugation to English yoke. Effect of this on Welsh? Second invasion of Edward-English fleet sent to Anglesea, and Edward advances into the country by land. Retreat of Llewellyn towards the Wye. Welsh outflanked and defeated by Sir Edmund Mortimer, and Llewellyn slain, 1282. Conquest of country completed, 1283. Conquered land divided into counties, and placed under rule of Sheriffs (Note 3). Edward's eldest son born at Caernarvon, 1284, and received title of Prince of Wales (Note 4).

Expulsion of Jews, 1290.-Property of entire community confiscated, and sentence of banishment passed.

Jews not again found in England until time of Commonwealth.