466. A regular series of quotients is obtained, by dividing the difference of the powers of two quantities, by the difference of the quantities. Thus,

(y2 — a2) ÷ (y—a)=y+a,

(y 3 — a3) ÷ (y—a)=y2+ay+a2,

(y^ — a1) ÷ (y—a)=y3+ay2+a2y+a3,

(y3 — a3) ÷ (y—a)=y*+ay3 +a2y2+a3y+a*,

&c.

Here it will be seen, that the index of y, in the first term of the quotient, is less by 1, than in the dividend; and that it decreases by 1, from the first term to the last but one:

While the index of a increases by 1, from the second term to the last, where it is less by 1, than in the dividend. This may be expressed in a general formula, thus,

(3TM —am(÷ (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.

So mult. ym-1+aym-2+a2ym3....+am-2y+am-1
Into

y-a

SECTION XVII.

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.)

(a+b)2=a2+2ab+b*

(a+b)3=a3+3a2b+3ab2+13

3

(a+b)=a*+4a3b+6a2b2+4ab3+bi

(a+b) =a3+5a*b+10a3b2 +10a2b3 +5ab*+b3, &c.

By attending to this series of powers, we shall find, that the exponents preserve an invariable order through the whole. This will be very obvious, if we take the exponents by themselves, unconnected with the letters to which they belong.

* Simpson's Algebra, Sec. 15. Simpson's Fluxions, Art. 99. EuJer'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.

Ff

In the square, the exponents

In the cube, the exponents

In the 4th power, the exponents

In the 5th power, the exponents

of a are 2, 1,
of b are 0, 1, 2
(of a are 3,2,1,0
of b are 0,1,2,3

of b are 0,1,2,3,4
of a are 4,3,2,1,0

of

a are 5,4,3,2,1,0

of b are 0,1,2,3,4,5, &c. Here it will be seen, at once, that the exponents, of a in the first term, and of b in the last, are each equal to the in-dex of the power; and that the sum of the exponents of the two letters is in every term the same. Thus in the fifth power,

The sum of the exponents

in the first term, is 5+0=5

It is farther to be observed,. that the exponents of a regu Larly decrease to 0, and that the exponents of b increase from 0. That this will universally be the case, to whatever extent the involution may be carried, will be evident, if we consider, that, in raising from any power to the next, each term is multiplied both by a and by b..

[of a in each term.

a3+2a2b+ab2, Here 1 is added to the exp. a2b+2ab2+b3, Here 1 is added to the [exp. of b in each term.

(a+b)3=a3+3a2b+3ab2+b3

If the exponents, before the multiplication, increase and decrease by 1, and if the multiplication adds 1 to each, it is evident they must still increase and decrease in the same manner as before.

469. If then a+b be raised to a power whose exponent

is n,

The exp's of a will be n, n−1, n—24.... 2, I, 0; And the exp's of b will be 0, 1, 2, ....n-2, n-1, n.

The terms in which a power is expressed, consist of the letters with their exponents, and the co-efficients. Setting

aside the co-efficients for the present, we can determine, from the preceding observations,the letters and exponents of any power whatever.

Thus the 8th power of a+b, when written without the co-efficients, is

· a® + a2 b + a ® b2 + a 3 b3 + a*b*+a3b3 +a2b® +ab+b3. And the nth power of a+b.is,

a2+an-1b+a22f2.... à2 b22+abn−1 +b2.

470. The number of terms is greater by 1, than the index. of the power. For, if the index of the power is n, a has, in different terms, every index from n down to 1; and there is one additional term which contains only b. Thus,

471. The next step is to find the co-efficients.

of the subject is more complicated.

This part

In the series of powers at the beginning of art. 468, the co-efficients, taken separate from the letters, are as follows;

The order which these co-efficients observe is not obvious, like that of the exponents, upon a bare inspection. But they will be found on examination to be all subject to the following law;

472. The co-efficient of the first term is 1; that of the second is equal to the index of the power; and universally, if the co-efficient of any term be multiplied by the index of the leading quantity in that term, and divided by the index of the following quantity increased by 1, it will give the co-efficient of the succeeding term.*

Of the two letters in a term, the first is called the leading quantity, and the other, the following quantity. In the examples which have been given in this section, a is the leading quantity, and b the following quantity.

It may frequently be convenient to represent the co-efficients, in the several terms, by the capital letters, A,B,C, &c.

* See Note Q,

The nth power of a+b, without the co-efficients, is

a” +an-1b+an-272+ar¬3f3+ar¬sf1, &c. (Art. 469.) And the co-efficients are,

An, the co-efficient of the second term;

The regular manner in which these co-efficients are derived one from another, will be readily perceived.

473. By recurring to the numbers in art. 471,it will be seen, that the co-efficients first increase, and then decrease at the same rate; so that they are equal, in the first term and the last, in the second and last but one, in the third and last but two; and, universally, in any two terms equally distant from the extremes. The reason of this is, that (a+b)" is the same as (b+a)"; and if the order of the terms in the binomial root be changed, the whole series of terms in the power will be inverted.

It is sufficient, then, to find the co-efficients of half the terms. These repeated, will serve for the whole.

474. In any power of (a+b), the sum of the co-efficients is equal to the number 2 raised to that power. See the list of co-efficients in art. 471. The reason of this is, that, according to the rules of multiplication, when any quantity is involved, the letters are multiplied into each other, and the co-efficients into each other. Now the co-efficients of a+b being 1+1=2, if these be involved, a series of the powers of 2 will be produced.