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433. The first member of an equation involving but one unknown quantity, to which all the terms have been transposed, is a polynomial in the most general sense of the term, and may be operated on as an algebraic quantity, without reference to the equation and to the particular values of the unknown quantity which will reduce the polynomial to zero, or satisfy the equation. If we take the polynomial,
Ax" + Bxm-' + Cem-to... + Sxo+Tx+U, and multiply each term by the exponent of x in that term, then diminish this exponent by unity, and form the algebraic sum of the results, we shall have
m Axn-'+(m—1Bxm-+(m—2) Cam-' to....+2 Sx+ T. Constructing a third polynomial from this, in the same way that this was derived from the first, we have m(m-1)Axm-' +
(m-1)(m—2)Bx*-+(m-2)(m—3) Com to...+25. A fourth may be formed from the third according to the same law; and so on, until we arrive at an expression which will be independent of x, because the degree, with respect to x, of any polynomial thus formed, is one less than of that which immediately precedes it.
Denoting the given polynomial by X, the second by Xį, the third by X2, etc., then
X, is the first derived polynomial of X,
X,, etc. And X1, X2, X, are the successive derived polynomials of X, and are called first, second, third, etc., derived polynomials.
Preserving the above notation, we have for the successive derived polynomials, the following
RULE.-To form X, multiply every term of X by the exponent of small x in the term, then diminish this exponent by unity and take the algebraic sum of the results. X2 is derived from X, in the same way that X, is derived from X; and so on.
What are the successive derived polynomials of
1st, 5.30* +4.5x'—3.9.co+2.7.0_8;
2d, 4.5.32° +3:4.5x*—2:3.93+2.7; Ans. 3d, 18.104.22.168*+22.214.171.124–2.3.9;
COMPOSITION OF DERIVED POLYNOMIALS.
434. Let us take the polynomial,
2"+2.37–+B.2"-+.... + Tc+U= X; and suppose
that its binomial factors of the first degree with respect to x are
X-a, x—b, x— C,.. Xm, xn. We shall then have the identical equation, 20* + Ax*-+.... + Tx+U= (x-a)(x-6)....(x-m)(x-n), which will subsist as a true equation, whatever quantity be substituted for x in its two members. Replace a by ytx; then (y+x)*+A(y+x)*+.. =(y+.c-a)(y+ič-6)..(y+i-n), in which the terms x—a, x—6, may be regarded as single, and hence the factors of the second member as binomial. Now the terms of the first member of this equation, developed and arranged with reference to the ascending powers of y, will give X+X/4+*y+..+
1.2..(m-1) And if the second member be developed, and arranged in the same manner, then by (424,5), the coefficient of yo will be
(x-a) (x—b)....(x-m)(x-n). The coefficient of y must be the algebraic sum of the products of the factors x--a, x—b, etc., taken m-1 in a set.
The coefficient of ymust be the algebraic sum of the products of these factors taken m-2 in a set.
In short, these coefficients may all be formed according to the law which governs the product of any number of binomial factors. .
But the coefficients of the like powers of y in these two developments must be equal; (368, III). Hence,
X= (x—a)(x-6)(—c)....(x—m)(x-1); and since the sum of all the products that can be formed by multiplying m factors in sets of me
1-1 and m—1, is the same as the sum of all the quotients which can be obtained by dividing the continued product of the factors, by each factor separately, it follows that X X
ti .+ + So likewise the sum of the products of the binomial factors taken m—2 and m2, is the same as the sum of all the quotients obtained by dividing the continued product by all the different products of the binomial factors taken 2 and 2; that is, X X
t. + 2 (x-a)(x-6) (x—a)(x—c)
(2-a)(x-n) By like reasoning it may be shown that X
; 2.3 (2-a)(—b)(x-c) (x--a)(x—m) n) and so for the next coefficient in order, etc., etc.
435. It has been seen (427) that if a, b, c,....,m, n are the roots of the equation,
X = 2 +A.xm-'+B.2-*+...+Tx+U= 0, it may be written,
X= (x—a)(x-6)(x—c)....(o—m)(x~n) = 0. Now if a number p of these roots are each equal to a, a number q equal to b, and a number r equal to c, the last equation becomes
X= (x—a)'(x—b)'(x–c)....(x—m)(x—n) = 0. But since X contains p factors equal to x—a, q factors equal to -6, r factors equal to x–c, its first derived polynomial will con
X tain the term p times, the term
9 times, the term -6
X X r times, besides the terms
etc., corresponding to the single roots, (434); that is, px qX rX
+ + + + +
X-m X n The factor (x—a) is found in every term of this expression for X, except the first, from which one of the p equal factors, xấa, has been suppressed by division. Hence, (x-a)p- is the highest power of
which is a factor common to all the terms of X,. For like reasons (x-6)-, (—c) are the highest powers of the factors x—b, x—c, which are common to all the terms of X1 ; hence,
(x—a)2-1(2—6)2-1(3-c)"1, is the greatest common divisor which exists between the first member of the proposed equation and its first derived polynomial.
The supposition that the given equation contains one or more sets or species of equal roots, necessarily leads to the existence of this greatest common divisor. Conversely :- if there be a common divisor between X and X, there must be one or more sets of equal roots belonging to the equation.
For, if (x—a)' be a factor of the greatest common divisor, then the composition of X, shows that (x-a)'+1 is a factor of X, and that a is therefore t+1 times a root of the equation X= 0. Hence the conclusions :
1.-An equation involving but one unknown quantity, x, and of which the second member is zero, has equal roots if there be between its first member, X, and its first derived polynomial X1, a common divisor containing x.
2.—The greatest common divisor, D, of X and X1, is the product of those binomial factors of X, of the first degree with respect to x, which correspond to the equal roots, each raised to a power whose exponent is one less than that with which it enters X. Therefore,
To determine whether an equation has equal roots, and if so, to find them, if possible, we have the following
RULE.—I. Seek the greatest common divisor between the first member of the proposed equation and its first derived polynomial,
If no common divisor be found, there are no equal roots; but if one be found, there are equal roots; in which case
II. Make an equation by placing the greatest common devisor, D, equal to zero; then any quantity which is once a root of D=0 will be twice a root of X = 0; any quantity which is twice a root of D=0 will be three times the root of X=0; and so on.
It will at once be seen that, if D contains a factor of the form (x—a)', t being a positive whole number greater than unity,
and we denote the greatest common divisor which exists between D and its first derived polynomial D1, by D', then D' will contain the factor (x—a)-1. And, again, denoting by D', the first derived polynomial of D', and by D' their greatest common divisor, (x-a)–? will be a factor of D". This process being continued, as the exponent of (c—a),—and consequently, the degree of the greatest common divisor,—diminishes by one for each operation, it is plain that when the degree of the equation,
D 0, is too high to be solved, we may in certain cases make the determination of the equal roots depend upon the solution of equations of lower degrees, until finally one is obtained which can be solved. To illustrate, suppose that for the equation,
X=0, it is found that
D= (–a)"(x-6)"(x–c); then
D' = (x-a)-1(x--6-1,
(x/a)"-"(3-6)"-", Din-1) = (2-a)(2-—b).
) The equation,
D(n-1) = (-a)(x—b) = 0, may be solved, giving the roots x= a, x=b, and
(-a)+1, (x-6)*+1, (x-c)', are factors of X, or a and b are each n+1 times roots, and c twice a root, of the equation,
X=0. Dividing the given equation by the product,
(3—()*+1(x-6)*+1(x—c)", its degree will be depressed 2n+4 units.