This may be expressed in a general formula, thus,

(yTM—aTM)÷(y—a)=ym-1+aym-2....+am¬2y+am−1,

To demonstrate this, we have only to multiply the quotient into the divisor. (Art. 115.)

All the terms except two, in the partial products, will be balanced by each other; and will leave the general product the same as the dividend.

Mult. y+ay3+a3y2+a3y+aa

466.b. In the same manner it may be proved, that the difference of the powers of two quantities, if the index is an even number, is divisible by the sum of the quantities. That is, as the double of every number is even;

(y2m—a2m)÷(y+a)=y2m—1— ay2m—2. +a2m-2y-a2m-1

....

And the sum of the powers of two quantities, if the index is an odd number, is divisible by the sum of the quantities. That is, as 2m+1 is an odd number;

(y2m+1+a2m+1) ÷ (y+a)=y2m—ay2m-1

For, in each of these cases, the product of the quotient. and divisor, is equal to the dividend.

Thus,

(y2-a2)÷(y+a)=y—a,

(ya—a1)÷ (y+a)=y3 — ay2+a2y—a3,

(y° —a®)÷ (y+a)=y3 — ay1 +a2y3 —a3y3+a*y—a3, &c.

And,

2

(y3+a3)÷(y+a)=y3—ay+aˆ,

(y3+a3)÷(y+a)=y*—ay3 +a3y” —a3y+a2,

(y1+a1)÷(y+a)=yo — ay3 +a3î1—a3y3+aay3—a3y+a®, &c.

GREATEST COMMON MEASURE.

466.c. The Greatest Common Measure of two quantities, may be found by the following rule;

DIVIDE ONE OF THE QUANTITIES BY THE other, and THE PRECEDING DIVISOR BY THE LAST REMAINDER, TILL nothing REMAINS; THE LAST DIVISOR WILL BE THE GREATEST COMMON MEASURE.

The algebraic letters are here supposed to stand for whole numbers. In the demonstration of the rule, the following principles must be admitted.

1. Any quantity measures itself, the quotient being 1.

2. If two quantities are respectively measured by a third, their sum or difference, is measured by that third quantity. If b and c are each measured by d, it is evident that b+c, and b-c are measured by d. Connecting them by the sign + or, does not affect their capacity of being measured by d.

Hence, if b is measured by d, then, by the preceding proposition, b+d is measured by d.

3. If one quantity is measured by another, any multiple of the former is measured by the latter. If b is measured by d, it is evident that b+b, 36, 4b, nb, &c. are measured by d.

Now let D=the greater, and d=the less of two algebraic quantities, whether simple or compound. And let the process of dividing, according to the rule, be as follows:

d)D(q
dq

In which q, q', q", are the quotients, from the successive divisions; and r, r', and o, the remainders. And as the dividend is equal to the product of the divisor and quotient added to the remainder,

Then, as the last divisor 'measures r, the remainder being o, it measures (2, and 3.) rq'+r'=d,

and measures

dq+r=D,

That is, the last divisor r' is a common measure of the two given quantities D and d.

It is also their greatest common measure. For every common measure of D and d, is also (3, and 2) a measure of D-dq=r; and every common measure of d and r, is also a measure of d-rq'=r'. But the greatest measure of r' is itself. This, then, is the greatest common measure of D

and d.

The demonstration will be substantially the same, whatever be the number of successive divisions, if the operation be continued, till the remainder is nothing.

To find the greatest common measure of three quantities; first find the greatest common measure of two of them, and then, the greatest common measure of this and the third quantity. If the greatest common measure of D and d be ', the greatest common measure r' and c, is the greatest common measure of the three quantities D, d, and c. For every measure of r', is a measure of D and d; therefore the greatest common measure of and c, is also the greatest common measure of D, d, and c.

The rule may be extended to any number of quantities.

446.d. There is not much occasion for the preceding operations, in finding the greatest common measure of simple algebraic quantities. For this purpose, a glance of the eye will generally be sufficient. In the application of the rule to compound quantities, it will frequently be expedient to reduce the divisor, or enlarge the dividend, in conformity with the following principle;

The greatest common measure of two quantities is not altered, by multiplying or dividing either of them, by any quantity which is not a divisor of the other, and which contains no factor which is a divisor of the other.

The common measure of ab and ac is a. If either be multiplied by d, the common measure of abd and ac, or of

moll

ab and acd, is still a. On the other hand, if ab and acd are the given quantities, the common measure is a; and if acd be divided by d, the common measure of ab and ac is a.

Hence, in finding the common measure by division, the divisor may often be rendered more simple, by dividing it by some quantity, which does not contain a divisor of the dividend. Or the dividend may be multiplied by a factor, which does not contain a measure of the divisor.

Ex. 1. Find the greatest common measure of 6a2+11ax+3x2, and 6a2 +7ax-3x2.

6a2+7ax-3x2)6a2+11ax+3x2(1

After the first division here, the remainder is divided by 2x, which reduces it to 2a+3x. The division of the preceding divisor by this, leaves no remainder. Therefore 2a+3x is the common measure required.

3

2. What is the greatest common measure of x3-b2 x, and x2+2bx+b2 ? Ans. x+b.

3. What is the greatest common measure of cx+x2, and a2c+a2x? Ans. c+x.

4. What is the greatest common measure of 3x3 — 24x−9, and 2x3-16x-6? Ans. x38x-3. 5. What is the greatest common measure of aa-ba, 4. and a5-b2 a3 ? Ans. a-b2.

6. What is the greatest common measure of 2 — 1; and xy+y? Ans. x+1.

7. What is the greatest common measure of 3-a3, and 24-a4?

8. What is the greatest common measure of a2-ab-2b3, and a23ab+2b2?

9. What is the greatest common measure of a' —xa, a3-a2x-ax2+x3 ?

and

10. What is the greatest common measure of a3 — ab2, a2+2ab+b2?

and

INVOLUTION AND EXPANSION OF BINOMIALS.*

ART. 467. THE manner in which a binomial, as well as any other compound quantity, may be involved by repeated multiplications, has been shown in the section on powers. (Art. 213.) But when a high power is required, the operation becomes long and tedious.

This has led mathematicians to seek for some general principle, by which the involution may be more easily and expeditiously performed. We are chiefly indebted to Sir Isaac Newton for the method which is now in common use. It is founded on what is called the Binomial Theorem, the invention of which was deemed of such importance to mathematical investigation, that it is engraved on his monument in Westminster Abbey.

468. If the binomial root be a+b, we may obtain, by multiplication, the following powers. (Art. 213.)

* Simpson's Algebra, Sec. 15. Simpson's Fluxions, Art. 99. Euler's Algebra, Sec. 2. Chap. 10. Manning's Algebra. Saunderson's Algebra, Art. 380. Vince's Fluxions, Art. 33. Waring's Med. Anal. p. 415. Lacroix's Algebra, Art. 135. Do. Comp. Art. 70. Lond. Phil. Trans. 1795, 1816, and 1817. Woodhouse's Analytical Calculation.