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b; and one
Here are the two positive roots, viz, x = a, x = negative root, viz, ac: the equation being constituted of the continued product of the three factors,
a = 0,
From an inspection of the equations A, B, 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. Indeed, 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 equation be a-(2a-2c).x3 + (a2+b2 - 4ac + c2 + d2).x2 + (2a2c + 2b2c - 2ac2 — 2ad2)x + (a2 + b2). (c2 +- d2) = 0. This may be regarded as constituted of the two subjoined quadratic equations, x2-2ax+a2 + b2 = 0, x2 + 2cx + c2 + d2 = 0 ; and each of these quadratics contains two imaginary roots; the first giving x = a ± b√ −1, and the second x ==c± dv- 1.
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 quan tities, 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.
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 coefficient equal to the sum of all the products which can be formed by multiplying all the roots two and two.
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 product 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. second equation of the same article, which has its roots partly real, partly imaginary, will want the second term if b− a + 2c = 0, or a b2c.
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 + r = 0, may be written x3 + 0 x2 + 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= 0, which has three positive roots 1, 2, and 5, we write x3 + 8x2 + 17x + 10 = O, this latter equation will have three negative roots x = 1, x =
5. In like manner, if instead of the equation 3+ 2x2-13x+10=0, which has two positive roots x=1, x=2, and one negative root = - 5, there be taken 3. 13x 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 − a) × (x −b)× (x−c)× (x− d) × &c = 0, and (x + a) × (x + b) 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 x” + ƒxm−1 + gxm−2+ hxm-3 + &c = 0, 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 = -f; 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 terin made less by twice the coefficient of the third term: viz, a2 + b2 + c2 + &c=f2g. 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 + bc + &c). But it is obvious, from equa. A, B, VOL. III.
that (a+b+6+ &c)2 = ƒ2, and (ab 4 ac + ic+ &c) = g. Thus we have ƒ2 = (a2 + b2 + c2 + &c) + 2g; and consequently a + b + c2 + &c = ƒ3 — 2g.
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 coefficient of the second term, and 3 times the coefficient of the fourth term: viz, a3 + b3 + c3 + &c - ƒ3 + 3ƒg — 3h. For we shall by actual involution, have (a + b + c + &c)3 = a3 + b3 + c3 + &c + 3(a + b + c) x (ab + ac + bc) — 3abc. But (a+b+c+&c)3 ~ ƒ3, (a + b + c + &c) x (ab + ac + bc + &ç)=−ƒg, abch. Hence therefore, − ƒ3 = a3 + b3 + c3 + &c 3fg3h; and consequently, a3 + b3 + c3 + &c = −ƒ3 + 3fg3h. And so on, for other powers of the roots..
In Every Equation, which contains only Real Roots:
I. If all the roots are positive, the terms of the equation will be and alternately.
II. If all the roots are negative, all the terms will have the sign +.
III. If the roots are partly positive, Dartly negative, there will be as many positive roots as there are variations of signs, and as many negative roots as there are permanencies of signs; these variations and permanencies being observed from one term to the following through the whole extent of the equation.
In all these, either the equations are complete in their terms, or they are made so.
The first part of this theorem is evident from the examination of equation A; and the second from equation B.
To demonstrate the third, we revert to the equation c (art. 3), which has two positive roots, and one negative. It may happen that either c > a + b, or c < a + b.
In the first case, the second term is positive, and the third is negative; because, having c> a+b, we shall have ac + be > (a + b) > ab. And, as the last term is positive, we see that from the first to the second there is a permanence of signs; from the second to the third a variation of signs; and from the third to the fourth another variation of signs. Thus there are two variations and one permanence of signs; that is, as many variations as there are positive roots, and as many permanencies as there are negative roots.
In the second case, the second term of the equation is negative, and the third may be either positive or negative. If
that term is positive, there will be from the first to the second a variation of signs; from the second to the third another variation; from the third to the fourth a permanence; making in all two variations and one permanence of signs. If the third term be negative; there will be one variation of signs from the first to the second; one permanence from the second to the third; and one variation from the third to the fourth: thus making again two variations and one permanence. The number of variations of signs therefore, in this case as well as in the former, is the same as that of the positive roots; and the number of permanencies, the same as that of the negative
Corol. Whence it follows, that if it be known, by any means whatever, that an equation contains only real roots, it is also known how many of them are positive, and how many negative. Suppose, for example, it be known that, in the equation .rs +3.x4 1200, all the roots are real: it may immediately be concluded that there are three positive and two negative roots. In fact this equation has the three positive roots = 1, x = 2, x = 3 ; and two negative roots, x =
- 27x2 + 166x
4, x =
If the equation were incomplete, the absent terms must be supplied by adopting cyphers for coefficients, and those terms must be marked with the ambiguous sign. Thus, if the equation were 205
20x3 + 30x2 + 19x
30 = 0,
all the roots being real, and the second term wanting. It must be written thus:
Then it will be seen that, whether the second term be positive or negative, there will be 3 variations and 2 permanencies of signs and consequently the equation has 3 positive and 2 negative roots. The roots in fact are, 1, 2, 3, −1, −5.
This rule only obtains with regard to equations whose roots are real. If, for example, it were inferred that, because the equation x2 + 2x + 50 had two permanencies of signs, it had two negative roots, the conclusion would be erroneous; for both the roots of this equation are imaginary.
Every Equation may be Transformed into Another whose Roots shall be Greater or Less by a Given Quantity. In any equation whatever, of which is unknown, (the equations A, B, c, for example) make x = z+m, z being a new unknown quantity, m any given quantitý, positive or negative: