180. To form the quotient we have only to substitute for the

the quotients, which are obtained by dividing these quantities by x-a, and which are respectively

Arranging the result with reference to the powers of x, we have

181. It is evident from the rules of division simply, that if the first member of the equation,

x2 + Px2-1+ Qx12 + &c. = 0,

be divided by x-a, the quotient obtained will be exhibited under the following form,

x2-1 + P1 x2-2 + Q 20^-3 + &c.

P', Q, &c. representing known quantities different from P, Q, &c. we have then

x2 + Pn2-1 + &c. = (x - a) (xn−1 + P' xn−2 + &c.); and according to what was observed in art. 116, the proposed equation may be verified in two ways, namely, by making x a = 0, or xn-1+P'xn-2 + &c. = 0.

Now if the equation

xn−1 + P2x2-2 + &c. = 0

has a root b, its first member will be divisible by x have then

xn−1 + P' x2=2` + &c. = (x — b) (xn−2 + P" x2-3 + &c.), and, consequently,

x2 + Pxn−1+ &c. = (x — a) (x — b) (xn−2 + P" xn−3 + &c.); the equation proposed may, therefore, be verified in three ways, namely, by making

x-α= 0, or xb0, or x2-2 + P3 + &c. = 0.

If the last of these equations has a root, c, its first member may still be decomposed into two factors,

= (x — a) (x — b) (x — c) (xn−3 + Pil x^-^ + &c.); from which it is obvious, that the proposed equation may be verified in four ways, namely, by making

x—a = 0, x — b — 0, x — - c = 0, x2¬3 + Px + &c. = 0. Pursuing the same reasoning, we obtain successively factors of the degrees

and if each of these factors being put equal to zero, is susceptible of a root, the first member of the proposed equation is reduced to the form

(x − a) (x — b) (x — c) (x — d) . . . . . (x — 1),

...

that is, it is decomposed into as many factors of the first degree, as there are units in the exponent, n, which denotes the degree of the equation.

may be verified in n ways, namely, by making

x-α=0, or x-b0, or xc0, or x-d=0,

It is necessary to observe, that these equations are to be regarded as true only when taken one after the other, and there arise manifest contradictions from the supposition, that they are true at the same time. In fact, from the equation x we obtain x = a, while x - b=0 gives x = b, results, which are inconsistent, when a and b are unequal quantities. 182. If the first member of the proposed equation,

x2 + P x2-1 + &c. = 0,

be decomposed into n factors of the first degree,

d,.... х

a = 0,

it cannnot be divided by any other expression of this degree. Indeed, if it were possible to divide it by a binomial x- a, different from the former ones, we should have

x2 + Pxn−1 + &c. = (x — α) (xn−1 + pxn-2 + &c.) and, consequently,

The second member vanishes by means of the factor α- α, which is nothing; this is not the case with respect to the first, which is the product of factors, all of which are different from zero, so long as a differs from the several roots a, b, c, d . . . l. The supposition we have made then is not true; therefore, an equation of any degree whatever does not admit of more binomial divisors of the first degree, than there are units in the exponent denoting its degree, and consequently, cannot have a greater number of roots.*

183. An equation regarded as the product of a number of factors,

equal to the exponent of its degree, may take the form of the product exhibited in art. 135, with this modification, that the terms will be alternately positive and negative.

If we take four factors, for example, we have

being the roots of the equation, taken with the contrary sign, the properties enumerated in art. 135, and proved generally in art. 136, will, in the present case, be as follows,

The coefficient of the second term, taken with the contrary sign, will be the sum of the roots;

The coefficient of the third term will be the sum of the products of the roots, taken two and two;

The coefficient of the fourth term, taken with the contrary sign, will be the sum of the products of the roots, multiplied three and

*This demonstration is taken from the Annales de Mathématiques published by M. Gergonne. See vol. iv, pp. 209, 210, note.

three, and so on, the signs of the coefficients of the even terms being changed;

The last term, subject also to this law, will be the product of all

the roots.

Making, for example, the product of the three factors

X- 5, x + 4, x + 3,

equal to zero, we form the equation

x3 + 2x2 22 x 60 = 0,

the roots of which are

we have for their sum

+5,4,3;

for the sum of their products, taken two and two, +5X-4+5X-3-4x-3=-20-15+12=23,

and for the product of the three roots,

the signs of those for the second and fourth terms.

If we make the product of the factors

60, changing

as it has no term involving 2, the power immediately inferior to that of the first term, wants the second term; and the reason is, that the sum of the roots, which, taken with the contrary sign, forms the coefficient of this term, is here

or zero, or in other words, the sum of the positive roots is equal to that of the negative.*

184. We have proved (182), that an equation, considered as arising from the product of several simple factors, or factors of the first degree, can contain only as many of these factors, as there are units in the exponent n denoting the degree of this equation; but if we combine these factors two and two, we form quantities of the second degree, which will also be factors of the proposed equation, the number of which will be expressed by

For example, the first member of the equation

xax3 abx2-abcx + a b c d = 0
-abcx+ab

+

(x — a) × (x — b) × (x — c) x (x — d),

may be decomposed into factors of the second degree, in the six following ways;

By combining the simple factors three and three, we form quantities of the third degree for divisors of the proposed equation; for an equation of the degree n the number will be

Of Elimination among Equations exceeding the First Degree.

185. THE rule given in art. 78, or the method pointed out in art. 84, is sufficient, in all cases, for eliminating in two equations an unknown quantity, which does not exceed the first degree, whatever may be the degree of the others; and the rule of art. 78, is applicable, even when the unknown quantity is of the first degree in only one of the proposed equations.

If we have, for example, the equations

ax2 + bxy + cy2 = m2,

x2 + xy = n2,

taking, in the second, the value of y, which will be