From these products, obtained by the common rule for algebraic multiplication, we discover the following laws:

1st. With respect to the exponents; the exponent of x, in the first term, is equal to the number of binomial factors employed. In the following terms, this exponent diminishes by unity to the last term, where it is 0.

2d. With respect to the co-efficients of the different powers of x: that of the first term is unity; the co-efficient of the second term is equal to the sum of the second terms of the binomials; the co-effi. cient of the third term is equal to the sum of the products of the different second terms taken two and two; the co-efficient of the fourth term is equal to the sum of their different products taken three and three. Reasoning from analogy, we may conclude that the co-efficient of the term which has n terms before it, is equal to the sum of the different products of the m second terms of the bi. nomials taken n and n. The last term is equal to the continued product of the second terms of the binomials.

In order to be certain that this law of composition is general, suppose that it has been proved to be true for a number m of binomials; let us see if it be true when a new factor is introduced into the product.

to be the product of m binomial factors, Na representing the term which has n terms before it, and Mam-n+1 that which immedi. diately precedes.

Let x+K be the new factor, the product when arranged according to the powers of X, will be

From which we perceive that the law of the exponents is evident

ly the same.

With respect to the co-efficients, 1st. That of the first term is

unity. 2d. A+K, or the co-efficient of a", is also the sum of the second terms of the m+1 binomials.

3d. B is by hypothesis the sum of the different products of the second terms of the m binomials, and A.K expresses the sum of the products of each of the second terms of the m first binomials, by the new second term K; therefore B+AK is the sum of the dif. ferent products of the second terms of the m+1 binomials, taken two and two.

In general, since N expresses the sum of the products of the second terms of the m first binomials, taken n and n; and as MK represents the sum of the products of these second terms, taken n— —1 and n-1, multiplied by the new second term K, it follows that N+MK, or the co-efficient of the term which has n terms before it, is equal to the sum of the different products of the second terms of the m+1 binomials, taken n and n. The last term is equal to the continued product of the m+1 second terms.

Therefore, the law of composition, supposed true for a number m of binomial factors, is also true for a number denoted by m+1. It is therefore general.

A

Let us suppose, that in the product resulting from the multiplication of the m binomial factors, x+a, x+b, x+c, x+d... we make a=b=c=d..., the indicated expression of this product, (x+a) (x+b)(x+c), will be changed into (x+a)". With respect to its development, the co-efficients being a+b+c+d..., ab+ac+ad+. abc+abd+acd . . ., the co-efficient of am-1, or a+b+c+d. becomes a+a+a+a+ . . ., that is, a taken as many times as there are letters a, b, c and is therefore equal to ma. The co-effi. cient of x-2, or ab+ac+ad+ . . ., reduces to a3+a2+a2. to a2 taken as many times as we can form different combinations with

m letters, taken two and two, or to m.

m -1 2

--a2. (Art. 163).

ór

The co-efficient of x-3 reduces to the product of a3, multiplied

by the number of different combinations of m letters, taken 3 and

3, or to

m

m-1 m- -2
2

a3, &c.

3

In general, if the term, which has n terms before it, is denoted by Nam, the co-efficient, which in the hypothesis of the second terms being different, is equal to the sum of their products, taken n and n, reduces, when all of the terms are supposed equal, to a" multiplied by the number of different combinations that can be made with m letters, taken n and n. Therefore

165. By inspecting the different terms of this development, a simple law will be perceived, by means of which the co-efficient of any term is formed from the co-efficient of the preceding term.

The co-efficient of any term is formed by multiplying the co-efficient of the preceding term by the exponent of x in that term, and dividing the product by the number of terms which precede the required

term.

For, take the general term

P(m-n+1)
Qxn

-anxm-n

This is called

the general term, because by making n=2, 3, 4..., all of the others can be deduced from it. The term which immediately pre

ber of combinations of m letters taken n-1 and n- -1. Here we

see that the co-efficient

P(m-n+1)
Qx n

is equal to the co-efficient

P

Q

which precedes it, multiplied by m-n+1, the exponent of x in

that term, and divided by n, the number of terms preceding the required term. This law serves to develop a particular power, with. out our being obliged to have recourse to the general formula. For example, let it be required to develop (x+a). From this law we have,

(x+a)=x+6ax2+15a2x2+20a3x2+15a2x2+6a5x+a®.

m-1

After having formed the first two terms from the terms of the general formula x+max"1+. . . ., multiply 6, the co-efficient of the second term, by 5, the exponent of x in this term, then divide the product by 2, which gives 15 for the co-efficient of the third term. To obtain that of the fourth, multiply 15 by 4, the exponent of x in the third term, and divide the product by 3, the number of terms which precede the fourth, this gives 20; and the co-efficients of the other terms are found in the same way.

In like manner we find

(x+a)1o=x1o+10ax2+45a2x2+120a3x7+210a1x®,

+252a5x5+210a x2+120a7x3+45a3x2+10a3x+a1o.

166. It frequently occurs that the terms of the binomial are affected with co-efficients and exponents, as in the following example. Let it be required to raise the binomial 3a2c-2bd to the 4th power.

Placing 3a2c=x and -2bd=y, we have

(x+y)=x+4x3y+6x3y2+4xy3+yʻ.

Substituting for x and y their values, we have

(3a3c—2bd)1= (3a2c)*+4(3a2c)3( — 2bd)+6(3a2c)2(—2bd)2+ 4(3a c) (-2bd)3+(−2bd)*,

or, by performing the operations indicated

(3a3c-2bd)=81a3c1-216a°c3bd+216a1c2b3d2—96a2cb3d3
+16b*d*.

The terms of the development are alternately plus and minus, as they should be, since the second term is —.

167. The powers of any polynomial may easily be found by the binomial theorem.

For example, raise a+b+c to the third power.

First, put
Then

b+c=d.

(a+b+c)3=(a+d)3=a3+3a2d+3ad+d3.

Or, by substituting for the value of d,

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

3a2c+3b2c+6abc

+3ac2+3bc2
+ c3.

This expression is composed of the cubes of the three terms, plus three times the square of each term by the first powers of the two others, plus six times the product of all three terms. It is easily proved that this law is true for any polynomial.

To apply the preceding formula to the development of the cube of a trinomial, in which the terms are affected with co-efficients and exponents, designate each term by a single letter, then replace the letters introduced, by their values, and perform the operations indicated. From this rule, we will find that

(2a2-4ab+362)3-8a-48a5b+132a4b2-208a3b3

+198a2b-108ab5+27bo.

The fourth, fifth, &c. powers of any polynomial can be developed in a similar manner.

Consequences of the Binomial Formula.

168. First. The expression (x+a)m being such, that x may be substituted for a, and a for x, without altering its value, it follows that the same thing can be done in the development of it; therefore, if this development contains a term of the form Ka"am-n, it must have another equal to Kx"a"-" or Ka" "x". These two terms of the development are evidently at equal distances from the two extremes; for the number of terms which precede any term, being indicated by the exponent of a in that term, it follows that