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212. It is proposed to demonstrate, in the following articles, several useful principles, which on account of their difficulty, were omitted from the body of the work. The subjects embraced, are the principles used in factoring, the binomial formula for any exponent, and the summation of series.
Principles used in Factoring.
213. FIRST PRINCIPLE. The difference of the like powers of any two quantities, is divisible by the difference of the quantities.
To demonstrate this principle, let a and b denote any two quantities, and in any positive whole number; then will am - Bm denote the difference between the like powers of any two quantities, and a - b the difference between the quantities ; we commence the division by the rule, we shall have the following
the remainder may be factored, and placed under the form,
6am-1 - 1m-1); hence, we have,
The second member of equation (1) will be entire, and consequently, the first member will be entire, when
is entire : that is, if the difference of the - b (m – 1) powers of two quantities is divisible by the difference of the quantities, then will the difference of the mh powers of the two quantities also be divisible by the difference of the quantities.
But, we know that the difference of the second powers is divisible by the difference of the quantities; hence, from the principle above demonstrated, the difference of the cubes is also divisible by the difference of the quantities; it having been proved that the difference of the cubes is divisible, it follows, from the principle demonstrated, that the difference of the fourth powers is also divisible by the difference of the quantities; the difference of the fourth powers being divisible, it follows, as before, that the difference of the fifth powers is divisible; and so on, by successive deduction, it may be shown that the division is possible when m is any positive number whatever; hence, the principle is proved.
We found the first term of the quotient to be, am-1; and if we perform a second partial division, we shall get, for the second term of the quotient, am-25, with a second remainder, 72(am-2 – 3m–2); dividing again, we shall find for the the third term of the quotient am-32; and so on. Writing out the quotient, we have,
= am-' + am-2b + am-362+...+abm-2 + 6m-1. (2).
The coefficient of each term of the quotient is equal to 1, and the exponents follow the law explained in deducing the binomial formula (Art. 100).
SECOND PRINCIPLE. The difference of like even powers of any two quantities is divisible by the difference of the squares of the quantities.
For, if we replace a byc? and b by da, equation (2) becomes,
Whatever may be the value of m, 2m is an number, and the second member is entire; hence, the principle is proved.
THIRD PRINCIPLE. The difference of like even powers of any two quantities, is divisible by the sum of the quantities.
For, if we multiply both members of (3) by the quantity (c – d), and reduce, we have,
-d) (020-2 + 2m-4d2 + ... + d2m-2). (4).
The second member of (4) is entire; hence, the principle is demonstrated.
FOURTH PRINCIPLE. The difference of the like powers of two quantities, is divisible by the difference of any other like powers of the two quantities, if the exponent in the first case is divisible by that in the second case.
If we replace d, in equation (2), by c", and b by M", n being a whole number, we have,
The second member of (5) is always entire; hence, the principle is demonstrated.
FIFTH PRINCIPLE. The sum of like odd powers of any two quantities is divisible by the sum of the quantities.
Let m be any odd number, and let the operation of dividing am + 7" by a + b be commenced as shown below : am + 6m
; am + am–10 am-1
- am-17 + 2m
factoring the remainder, and writing it over the divisor, as a fraction, we have,
a + 6
Since m is an odd number, m - 1 is an even number, and consequently the quantity within the brackets is entire, according to the third principle; hence, the proposition is proved.
The form of the quotient is the same as that of am – Em by a – b, except that the terms are alternately plus and minus ; that is,
am + bm
- abm-2 + 2-1
Value of (am – bm) = (a - b), when a = = b.
214. It has been shown that, when m is a positive whole number, we have,
a = b.
in which the quotient has m terms. This is true for all values of a and b; hence, it will be true when a"
0 In this case,
Ō sequence of the existence of a factor in both numerator and denominator, which becomes 0 under the supposition that a = b; denoting what the true value of this fraction becomes, when a = b, by the symbol, ram
and then making a = b in the second b member, we have,
It may be shown that equation (1) is true whatever may be the value of m; that is, whether m is positive or negative, entire or fractional.