CHAPTER III

FORMULA FOR THE EXPANSION OF A BINOMIAL

623. The product of two polynomials is equal to the sum of the products which are obtained by multiplying each term of the multiplicand by each term of the multiplier. In general, the product of several polynomials is the sum of the products which are obtained by taking in all possible ways a term in each of the given polynomials.

Suppose that it is desired to find the product of n binomial factors,

arranged with respect to the decreasing powers of x. According to the law which has been stated, the product of these binomials is the sum of the products which one obtains by taking in all possible ways a term from each of them. The first term of the product will be found by taking the product of the n first terms, that is, a". If one takes the second term a,, of the first binomial, and the first term x, of all the other binomials, he obtains the product ax"-1; similarly, on taking the second term a, of the second binomial with the first term of all the other binomials one has a-1; a second term of any of the binomials combined with the first term x of all the other binomials furnishes a term involving "-1; if one adds together all the terms of the degree n 1, he sees that the coefficient of xn-1 is

a + a + a ++a-1+ an

which, for brevity, is called S. Hence, the second term of the product is S-1.

Form now the products of the two second terms of any two binomials with the first term x of all the remaining binomials; then the terms of the degree n 2 of the product will be obtained, such as

a2, aаx-2, aa"-2, etc.; on adding these terms together, it is seen that the coefficient of "-2 is the sum of the products of the quantities a, a a, taken two at a time, which is represented

by S. Hence the third term is S„x”−2.

n

On forming all the products of the second terms of any three binomials and the first term .c of all the other binomials, the terms of the degree 3 of the product are obtained, such as, aaa-3, à ̧aà ̧13, etc. On adding these terms together, and calling S the sum of the products .. a- a taken three at a time, the fourth term S"-3 of the product is obtained.

--

In general, on taking the second term of any r of the binomial factors and the first term x of the other remaining n > terms, the term of the degree n r is formed; on adding these terms together and calling S, the sum of the product of the quantities аp ag a, taken at a time, the general term S," of the product is obtained.

The term of the first degree will be found by forming the product of the second term in all the binomial factors, excepting one, with the first term x of this remaining factor, these terms added together will give the last term but one of the product, S-. The last term of the product required will be the product of the second terms of the binomial factors, namely a, a, Az

call Sn

which we

Hence the product of the n binomial factors is expressed as follows:

-2

x2 + S1x"¬1 + Sqx"2 + . . . + S„x”-” +

Suppose that the quantities a, az az...

a, then the product of the n factors

(x + a ̧) (x + a2)

+ Sn-1x + Sn.

a, are all equal to

(x + an)

takes the form (x+a)". Moreover, the sum S, of the quantities a, a,

a, is na, since each of these quantities, a in number, is equal to a. The symbol S, represents the sum of the products of these same quantities taken two at a time; every such product is a2, and the number of them is the number of the combinations of n things n(n−1) n(n−1)a2. taken two at a time, or ; therefore their sum is equal to 2! Similarly, S, designates the sum of the products of these same n quantities taken three at a time; since each of them is equal to a3

2!

a a

In general, S, represents the sum of the products of n quantities taken at a time; since each of the quantities is equal to a, each of the products is equal to a"; since their number is the number of combinations of n things taken r at a time,

Finally, the product of the quantities a, a,..... a, is a". Therefore the formula

r!

n

which is known as the binomial formula. It is of very great practical use, serving to form the development of any positive integral power of a binomial. It will be shown later that the law of expansion exhibited in formula (i) holds for any exponent. The general term which is called the (r+1)th step in the development is, as has been seen,

n(n − 1) . . . (n − r+1) a2x2-r.

r!

The development for (x-a)" is deduced from formula (i) on substituting a for a in (i), thus

624. Characteristics of Development of the Binomial (x+a)", where n is a Positive Integer.

1. The exponent of x in the first term is n, and decreases in each succeeding term by unity.

2. The exponent of a in the second term is one, and increases uniformly by unity.

3. The sum of the exponents of x and a in every term is the same and equal to n, the degree of the binomial.

4. The number of terms in the expansion of (x + a)" is n + 1; because the exponents of x form the series of the first n integral numbers plus the exponent zero of the last term,

The coefficients of the terms equally distant from the extreme terms are equal. The coefficient of the second term in the expansion of (a) in formula (i), 623, is C1, and of the third term C, etc.;

hence

n

n

Both the first and the last terms have the same coefficient, unity, the second and the term before the last have the coefficients „C and C-1; but from the theorem proved in 617, it follows that these two expressions are equal. Similarly, the third terms, counting from the extremities of the expansion, have as coefficients the equal numbers, C and C-2, etc.

n

The coefficients of the expansion for (x + a)" are connected by a very simple law: the coefficient of any term multiplied by the exponent of x in that term and divided by the number of the term will give the coefficient of the next term.

which, multiplied by the exponent (2) of x in that term and divided by 3, the number of the term, gives

which, multiplied by the exponent (n—r+1) of x and divided by the number of the term r, gives the coefficient of the (r+1)th term,

Suppose that we put x=1, and a=x in equation (3) 624, then (6) (1+x)" = 1 + nC1 x + nС2 ¿x2 + nСzx3 + · · ·nСn−1x2¬1 + x2.

1

2

3

625. It is important to be able to develop rapidly any power of a binomial. The following illustrations will much assist the calcu

lation.

1. (x+a)ˆ = x2 +6 x3a + 15 x1a2 + 20 x3a3 + 15 x2a1 +6 xa3 +a3. The coefficient of the third term is found by multiplying 6 by 5 and dividing by 2; the coefficient of the fourth term is 15 times 4

and the product divided by 3. Since the exponent of the binomial is 6, the number of terms in the expansion will be 6+1, and the coefficients of the remaining terms will be the same as those which precede the coefficient of the middle or fourth term in reverse order.

2. (x + a)o = x2 + 9 x3a + 36 x1 a2 + 84 xoa3 + 126 x3 a* + 126 x1a3 +84x3 a +36 x2 a2 + 9 xa3 + ao.

The development contains 10 terms; it is only necessary to calculate the first five terms; when the fifth term 126xa has been calculated, the coefficients are reproduced in reverse order.

3. (x — a)1o1o — 10.x9a + 45 x3a2 — 120 x1a3 + 210 xoa1—252 x3a3 +210 x1α- 120 x3 a2 + 45 x2a3 - 10 xao + a1o.

Since the number of terms is 10+1, which is odd, the last term 'will have the sign+, and the terms equally distant from the ends will have the same signs.

4. (x +462 x3a

- a)11— x11 —11 x1a + 55 x9a2-165 x3a3 +330 x1a1—462xoa3

=

- 330 x1a2 + 165 x3 a3 — 55 x2a3 +11 xa1o — a11.

Since the number of terms is even, the last term will have the sign and the terms which are equally distant from the ends will have contrary signs.

626. The coefficients increase from the beginning to the middle of the development, and diminish from the middle to the end.

We have already seen that the ratio of the (r + 1)th term to the 7th is

n Cr n−r+1.

nCr-1

[8624, (4), (5)]

The coefficients will continue to increase so long as the multiplier is greater than unity they begin, on the contrary, to decrease as soon as this multiplier is less than unity. If one puts

of the development; hence the terms increase from the first term till the middle of the series; after the middle term, i. e., after the inequality is reversed, the coefficients decrease.