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Consequences deduced from the preceding Principles.
319. Every equation of an odd degree, the co-efficients of which are real, has at least one real root affected with a sign contrary to that of its last term.
For, let xm+Fxm~1+ . . . Ta;±U=0, be the proposed equation; and first consider the case in which the last term is negative.
By making x=0, the first member becomes —U. But by giving a value to x equal to the greatest co-efficient plus unity, or (K+l), the first term xm will become greater than the arithmetical sum of all the others (Art. 314), the result of this substitution will therefore be positive; hence, there is at least one real root comprehended between 0 and K+l, which root is positive, and consequently affected with a sign contrary to that of the last term.
Suppose now that the last term is positive.
Making a;=0, we obtain +U for the result; but by putting — (K+l) in place of x, we shall obtain a negative result, since the first term becomes negative by this substitution; hence the equation has at least one real root comprehended between 0 and —(K+l,) which is negative, or affected with a sign contrary to that of the last term.
320. Every equation of an even degree, involving only real coefficients of which the last term is negative, has at least two real roots, one positive and the other negative. For, let —U be the last term; making x=0, there results —U. Now substitute either K+l, or —(K+l), K being the greatest co-efficient of the equation; as m is an even number, the first term xm will remain positive; besides, by these substitutions, it becomes greater than the sum of all the others; therefore the results obtained by these substitutions are both positive, or affected with a sign contrary to that given by the hypothesis x=O; hence the equation has at least two real roots, one comprehended between 0 and K+l, or positive, and the othei between Oand — (K+l), or negative.
321. If an equation, involving only real co-efficients, contains imaginary roots, the number of these roots must be even.
For, conceive that the first member has been divided by all the simple factors corresponding to the real roots ; the co-efficients of the quotient will be real (278); and the equation must also be of an even degree; for if it was uneven, by placing it equal to zero, we should obtain an equation that would contain at least one real root, which, from the nature of the equation, it cannot have.
Remark. 322. There is a property of the above polynomial quotient which belongs exclusively to equations containing only imaginary roots; viz. every such equation always remains positive for any real value substituted for x.
For, if it could become negative, since we could also obtain a positive result, by substituting K+l or the greatest negative co-efficient plus unity for x, it would follow that this polynomial placed equal to zero, would have at least one real root comprehended between K+l and the number which would give a negative result.
It also follows, that the last term of this polynomial must be posi. live, otherwise a;=0 would give a negative result.
323. When the last term of an equation is positive, the number of Us real positive roots is even; and when it is negative this number is uneven.
For, first suppose that the last term is +U, or positive. Since by making x=0, there will result +Q, and by making a;=K+l, the result will also be positive, it follows that 0 and K+l give two results affected with the same sign, and consequently (311), the number of real roots, (if any), comprehended between them, is even.
When the last term is — U, then 0 and K+l give two results affected with contrary signs, and consequently comprehend either a single real root, or an odd number of them.
The reciprocal of this proposition is evident.
324. An equation of any degree whatever cannot have a greater number of positive roots than there are variations in the signs of its terms, nor a greater number of negative roots than there are permanences of these signs.
In the equation x—a—0, there is one variation, that is a change of sign in passing along the terms, and one positive root, x=a. And in the equation x+6=0, there is one permanence, and one negative root, x=—b.
If these equations be multiplied together, there will result an equation of the second degree,
If o is less than b, the equation will be of the first form (Art. 144); and if a>6 the equation will be of the second form: that is
a<6 gives x*+px—^=0 and
In either case, there is one variation, and one permanence, and since in either form, one root is positive and one negative, it follows that there are as many positive roots as there are variations, and as many negative roots as there are permanences.
The proposition would evidently be demonstrated in a general
manner, if it were shown that the multiplication of the first member
by a factor x—^corresponding to a positive root, would introduce at
least one variation, and that the multiplication by a factor aj+a,
would introduce at least one permanence.
Let there be the equation
r±Ax'"-,±Ba«-!'±Caf-3± . . . ±TadbU=0,
in which the signs succeed each other in any manner whatever; by multiplying it by x—a, we have
a^-'iC a»-3± . . . ±U la;
The co-efficients which form the first horizontal line of this product, are those of the proposed equation, taken with the same sign; and the co-efficients of the second line are formed from those of the first, multiplied by a, taken with contrary signs, and advanced one rank towards the right.
Now, so long as each co-efficient of the upper line is greater than the corresponding one in the lower, it will determine the sign of the total co-efficient; hence, in this case there will be, from the first term to that preceding the last, inclusively, the same variations and the same permanences as in the proposed equation; but the last term zpUa having a sign contrary to that which immediately precedes it, there must be one or more variations than in the proposed equation.
When a co-efficient in the lower line is affected with a sign contrary to the one corresponding to it in the upper, and is also greater than this last, there is a change from a permanence of sign to a variation; for the sign of the term in which this happens, being the same as that of the inferior co-efficient, must be contrary to that of the preceding term, which has been supposed to be the same as that of its superior co-efficient. Hence, each time we descend from the upper to the lower line, in order to determine the sign, there is a variation which is not found in the proposed equation; and if, after passing into the. lower line, we continue in it throughout, we shall find for the remaining terms the same variations and the same permanences as in the proposed equation, since the co-efficients of this lino are all affected with signs contrary to those of the primitive co-efficients. This supposition would therefore give us one variation for each positive root. But if we ascend from the lower to the upper line, there may be either a variation or a permanence. But even by supposing that this passage produces permanences in all cases, since the last term zpUa forms a part of the lower line, it will bo necessary to go once more from the upper line to the lower, than from the lower to the upper. Hence the new equation must have at least one more variation than the proposed; and it will be the same for each positive root introduced into it.
It may be demonstrated, in an analogous manner, that the multiplication by a factor x+a, corresponding to a negative root, would introduce one permanence more. Hence, in any equation the number of positive roots cannot be greater than the number of VariaTions of sign, nor the number of negative roots greater than the number of Permanences.
325. Consequence. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences.
For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences; we shall have m~n+p. Moreover, let n' denote the number of positive roots, and p' the number of negative roots, we shall have m—n'-\-p'; whence
Now, we have just seen that n' cannot be >n, and p' cannot be >j>; therefore we must have n'=n, and p'=p.
Remark. 326. When an equation wants some of its terms, we can often discover the presence of imaginary roots, by means of the above rule.
For example, take the equation
p and q being essentially positive; introducing the term which is