ON THE NATURE AND SOLUTION OF EQUATIONS IN GENERAL.

1. In order to investigate the general properties of the higher equations, let there be assumed between an unknown quantity and given quantities a, b, c, d, an equation constituted of the continued product of uniform factors: thus (x-a) x (x-6) × (x−c) x (x−d) = 0. This, by performing the multiplications, and arranging the final product according to the powers or dimensions of x, becomes

Now it is obvious that the assemblage of terms which compose the first side of this equation may become equal to nothing in four different ways; namely, by supposing either x = a, or x=b, or xc, or x = d; for in either case one or other of the factors —a, ≈—b, x—c, x-d, will be equal to no. thing, and nothing multiplied by any quantity whatever will give nothing for the product. If any other value e be put for x, then none of the factors e—a, e-b, e- c, e-d, being equal to nothing, their continued product cannot be equal to nothing. There are therefore, in the proposed equation, four roots or values of x; and that which characterizes these roots is, that on substituting each of them successively instead of x, the aggregate of the terms of the equation vanishes by the opposition of the signs + and -.

The preceding equation is only of the fourth power or degree; but it is manifest that the above remark applies to equations of higher or lower dimensions: viz. that in general an equation of any degree whatever has as many roots as there are units in the exponent of the highest power of the unknown quantity, and that each root has the property of rendering, by its substitution in place of the unknown quantity, the aggregate of all the terms of the equation equal to nothing.

It must be observed that we cannot have all at once x=α, x = b,x= c, &c. for the roots of the equation; but that the b = 0, x-c= 0, &c. obtain only in a disjunctive sense. They exist as factors in

particular equations

a=0,x

the

the same equation, because algebra gives, by one and the same formula, not only the solution of the particular problem from which that formula may have originated, but also the solution of all problems which have similar conditions. The different roots of the equation satisfy the respective conditions: and those roots may differ from one another, by their quantity, and by their mode of existence.

It is true, we say frequently that the roots of an equation are x = a, x = b, x = c, &c. as though those values of x existed conjunctively; but this manner of speaking is an abs breviation, which it is necessary to understand in the sense explained above.

2. In the equation A. all the roots are positive; but if the factors which constitute the equation had been x+a, x + b, x+c, x+d, the roots would have been negative or subtractive.

Thus

x++ a x2 +ab x2 +abc x + abcd = 0.... (B)

+ac

+abú

has negative roots, those roots being x = 0, x — — 6, x=- - c, x == d: and here again we are to apply them disjunctively.

3. Some equations have their roots in part positive, in part negative. Such is the following:

x3

-b

+c

}

}

· +ab | 2

x + abc

--ac

-bc

=

=0

(C)

Here are the two positive roots, viz. x= a, x = b ; and one negative root, viz, xc: the equation being constituted of the continued product of the three factors, x — a = 0, x-b =0,x+ c = 0.

From an inspection of the equations A, a, c, it may be inferred, that a complete equation consists of a number of terms exceeding by unity the number of its roots.

4 The preceding equations have been considered as formed from equations of the first degree, and then each of them contains so many of those constituent equations as there are units in the exponent of its degree But an equation which exceeds the second dimension, may be considered as composed of one or more equations of the second degree, or of the third, &c. combined, if it be necessary, with equations of the first degree, in such manner, that the product of all those constituent equations shall form the proposed equation. In

deed

deed, when an equation is formed by the successive multiplication of several simple equations, quadratic equations. cubic equations, &c. are formed; which of course may be regarded as factors of the resulting equation.

5. It sometimes happens that an equation contains imaginary roots; and then they will be found also in its constituent equations This class of roots always enters an equation by pairs, because they may be considered as containing, in their expression at least, one even radical placed before a negative quantity, and because an even radical is necessarily preceded by the double sign ± Let, for example, the equa tion be x4-(2a — 2c)x 3 +(a2+b2 —4ac+c2+d2)x2+(2a2c+ 2b3c — 2ac2 — 2ad2)x+(a2+b2). (c2 + d2 ) = 0 This may be regarded as constituted of the two subjoined quadratic equations, 2 2ax+a2 + b2 = 0, x2+ 2cx+ c2 + d2 =0; and each of these quadratic contains two imaginary roots; the first giving x = a ± b √ − 1, and the second x → -c+ d√ -1.

3

-

In the equation resulting from the product of these two quadratics, the coefficients of the powers of the unknown quantity, and of the last term of the equation, are real quantities, though the constituent equations contain imaginary quantities; the reason is, that these latter disappear by means of addition and multiplication.

The same will take place in the equation (x—a). (x + b). (x2 + 2cx + c2 +d2) =0, which is formed of two equations of the first degree, and one equation of the second whose roots are imaginary.

These remarks being premised, the subsequent general theorems will be easily established.

THEOREM I.

Whatever be the Species of the Roots of an Equation, when the Equation is arranged according to the Powers of the Unknown Quantity, if the First Term be positive, and have unity for its Coefficient, the following Properties may

be traced :

I. The first term of the equation is the unknown quantity raised to the power denoted by the number of roots.

II. The second term contains the unknown quantity raised to a power less than the former by unity, with a coefficient equal to the sum of the roots taken with contrary signs.

III. The third term contains the unknown quantity raised to a power less by 2 than that of the first term, with a coeffi cient equal to the sum of all the products which can be formed by multiplying all the roots two and two.

Jan

ary

IV. The fourth term contains the unknown quantity raised to a power less by 3 than that of the first term, with a coefficient equal to the sum of all the products which can be made by multiplying any three of the roots with contrary signs.

V And so on to the last term, which is the continued pro.. duct of all the roots taken with contrary signs.

All this is evident from inspection of the equations exhibited in arts. 1, 2, 3, 5.

Cor. 1. Therefore an equation having all its roots real, but some positive, the others negative, will want its second term when the sum of the positive roots is equal to the sum of the negative roots. Thus, for example, the equation c will want its second term, if a + b = c.

Cor. 2. An equation whose roots are all imaginary, will want the second term, if the sum of the real quantities which enter into the expression of the roots, is partly positive, partly negative, and has the result reduced to nothing, the imaginary parts mutually destroying each other by addition in each pair of roots. Thus, the first equation of art. 5 will want the second term if-2a + 2c= 0, or a = c. The second equation of the same article, which has its roots partly real, partly imaginary, will want the second term if b − a + 2c = O, or a-b 2c.

Cor. 3. An equation will want its third term, if the sum of the products of the roots taken two and two, is partly positive, partly negative, and these mutually destroy each other.

Remark An incomplete equation may be thrown into the form of complete equations, by introducing, with the coefficient a cypher, the absent powers of the unknown quantity: thus, for the equation x3 + r0, may be written x3 +0x2 + 0 x + r = 0. This in some cases will be useful.

Cor. 4. An equation with positive roots may be transformed into another which shall have negative roots of the same value, and reciprocally. In order to this, it is only necessary to change the signs of the alternate terms, beginning with the second. Thus, for example, if instead of the equation x3-8x2 + 17x-10= O, which has three positive roots 1, 2, and 5, we write x3 + 8x2 + 17x + 10 = 0, this latter equation will have three negative roots x = x = -5. In like manner, if instead of the equation x3 + 2x2-13x+10=0, which has two positive roots x=1, x=2, and one negative root x = 5, there be taken x3

13.x

1, x

2x2

2,

10= 0, this latter equation will have two negative roots, x- 1,x=- 2, and one positive root x = 5. In general, if there be taken the two equations, (x {x—b) × (x−c) × (x−d) × &c = 0, and (x+a) ×(x+b)×

a) x

(x+

=

(x+c)× (x+d) × &c 0, of which the roots are the same in magnitude, but with different signs: if these equations be developed by actual multiplication, and the terms arranged according to the powers of x, as in arts. 1, 2; it will be seen that the second terms of the two equations will be affected with different signs, the third terms with like signs, the fourth terms with different signs, &c

When an equation has not all its terms, the deficient terms must be supplied by cyphers, before the preceding rule can be applied.

Cor. 5. The sum of the roots of an equation, the sum of their squares, the sum of their cubes, &c. may be found without knowing the roots themselves. For, let an equation of any degree or dimension, m, be xm + fxm-1 + gxm-2 + hxm-3 + &c O, its roots being a, b, c, d, &c. Then we

shall have,

1st. The sum of the first powers of the roots, that is, of the roots themselves, or a + b + c + &c = −ƒ; since the coefficient of the unknown quantity in the second term, is equal to the sum of the roots taken with different signs.

2dly. The sum of the squares of the roots, is equal to the square of the coefficient of the second term made less by twice the coefficient of the third term: viz, a2 + b2 + c2 + &c=f2 - 2g. For, if the polynomial a + b + c + &c. be squared, it will be found that the square contains the sum of the squares of the terms, a, b, c, &c. plus twice the sum of the products formed by multiplying two and two all the roots a, b, c, &c. That is, (a+b+c+ &c)2 = a2 + b2 + c2 + &c +2(ab + ac + be + &c). But it is obvious, from equa A, B, that (a+b+c+ &c)2 =ƒ2, and (ab + ac + bc + &c) = g, Thus we haveƒa = (a2 + b2 +c2 +&c) +2g; and consequently +62 + c2 + &c == ƒ2 — 28.

3dly. The sum of the cubes of the roots, is equal to 3 times the rectangle of the coefficient of the second and third terms made less by the cube of the co-efficient of the second term, and 3 times the coefficient, of the fourth term: viz. a3 +63 +c3 +&c=-ƒ3 + 3ƒg 3h. For we shall by actual involution, have (a + b + c + &c)3 = a3 +63 + c3 + &c + 3(a+b+c) x (ab + ac + bc) 3abc. But (a+b+c+&c)3 − ƒ3, (a + b + c + &c) × (ab + ac + bc + &c) abc h. Hence therefore, -ƒ3 ́a3 +63 + c3+ &c— 3fg+3h; and consequently, a3 + 63 + c3 + &c = −ƒ3 + 3fg 3h. And so on, for other powers of the roots.

=

THEOREM