But the coefficients of the like powers of y in these two developments must be equal (368, III). Hence,

X = (x − a) (x — b) (x — c) . . . . (x — m) (x — n) ;

and since the sum of all the products that can be formed by multiplying m factors in sets of m - 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, follows that

So likewise the sum of the products of the binomial factors taken m - 2 and m 2, 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,

and so for the next coefficient in order, etc., etc.

EQUAL ROOTS.

435. It has been seen (427) that if a, b, c,

the roots of the equation,

X:

...., m, n are

= x2 + Axm-1 + Bæm−2 + + Tx+U=0,

it may be written,

= 0.

X= (x − a) (x — b) (x — c) . . . . (x — m) (x — n) : 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)o (x — b)a (x — c)” . . . . (x m) (x — n) = 0. But since X contains p factors equal to x-a, q factors equal to x-b, r factors equal to x-c, its first derived polynomial will

responding to the single roots (434); that is,

+

+

The factor (a) is found in every term of this expression

for X, except the first, from which one of the p equal factors,

X

1

a, has been suppressed by division. Hence, (x — a)o–1 is the highest power of x -a, which is a factor common to all the

terms of X1.

For like reasons (x — b)a1, (x — c)-1 are the highest powers of the factors x b, x of X1; hence,

1

c, which are common to all the terms

(x — a)x−1 (x — b)a—1 (x — c)r—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 X1 there must be one or more sets of equal roots belonging to the equation.

1

1

For, if (xa) 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.

19

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 divisor, 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 D0 will be three times the root of X = 0; and so on.

1

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 (-a)-1. And, again, denoting by D', the first derived polynomial of D', and by D" their greatest common divisor, (x — a)12 will be a factor of D". This process being continued, as the exponent of (x-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,

it is found that then

The equation,

D(-1)=(xa) (x —b).

D(n−1) = (x − a) (x — b) = 0,

may be solved, giving the roots x = a, x = b, and

(x − a)n+1, (x·

a)+1, (x-b)+1, (x —c)2,

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,

(x — a)”+1 (x — b)n+1 (x — c)3,

its degree will be depressed 2n + 4 units.

equal roots, and if so, what are they?

The first derived polynomial of the first member is

The greatest common divisor between this and the first member of this equation is x — 2; therefore 2 is twice a root of the equa tion, and

24

2x3-7x2 + 20x - 12

may be divided twice by x-2, or once by (x-2)2 = x2—4x+4. Performing the division, we find the quotient to be 2+2x-3, and the original equation may now be written

(x2-4x+4) (x2 + 2x − 3) = 0.

This equation will be satisfied by the values of x found by placing each of these factors equal to zero. From the first we

get x =
2, x = 2, and from the second x = 1,
the four roots of the given equation are 1, 2, 2,

2. Find the equal roots of the equation

-3; hence

-=

3.

3. What are the equal roots of the equation
x5 — 2x4 + 323. 7x2+8x 3=0?

- 2x5 + 38x1 — 311⁄23 — 61x2 + 96x — 36=0?

30x5

Hence 1 is twice a root of the equation D= 0, and three times.

a root of the given equation.

Dividing D1-3x3-3x2+11x-6 by D2x2-2x+1, we find for the quotient 2x6 = (x − 3) (x + 2). Therefore, D= (x − 3) (x + 2) (x − 1)2,

and

436. Having an equation involving but one unknown quantity, to transform it into another, the roots of which shall differ from those of the proposed equation by a constant quantity.

Assume+Axm-1+Bxm2 + Cxm-3+....+Tx+U=0, and denote the new unknown quantity by y, and by x' the arbitrary but fixed difference which is to exist between the corresponding values of x and y; we shall then have x =

y + x'. Substituting this value of x in the given equation, it becomes (y + x')m + A (y+x')TM-1 + B (y+x') m−2 + C' (y+x')TM¬3 + . . . . + T(y + x) + U = 0.

m

Developing the terms separately, by the binomial formula, and arranging the aggregate of the results with reference to the ascending powers of y, we have

m-1

+ Ax'm-1

x'm-2

'm-2

2

+ Bx'm-2

+ Cx'm-3

+ B (m−2) x'
+ C(m-3) x'm-4

'm--3

+ A (m−1)

+ To'

+ B (m−2)

An examination of this developed first member leads to these conclusions:

1. The absolute term of the transformed equation, or the coefficient of yo, is what the first member of the given equation becomes when x' is substituted for x.