Ex. 7. Find the value of x in the equation ax+3= ‚2a*

+1.

1

Ans. x=

log, a

Ex. 8. Given log. x+log y={{ to find the values of x and log. z-log. y= and y.

Ex. 9. In the equation 210, it is required to find the value of x. Ans. x=3.321928, &c.

Ex. 10. Given /729-3, required the value of x.

Ans. x6.

Ex. 11. Given /57862=8, to find the value of x.

3

Ans. 5.2735, &c.

Ex. 12. Given (216)=64, to find the value of 2.

x

Ans. x 3.8774, &c.

Ex. 13. Given 43 =4096, to find the value of x.

Ex. 14. Given ax+y=c, and byd, to find the values of

320

CHAPTER XV.

ON

THE RESOLUTION OF EQUATIONS

OF THE THIRD AND HIGHER DEGREES.

§ 1. THEORY AND TRANSFORMATION OF EQUATIONS, 473. In addition to what has been already said (Art. 168), it may here be observed, that the roots of any equation are the numbers, which, when substituted for the unknown quantity, will make both sides of the equation identically equal. Or, which is the same, the roots of any equation are the numbers, which, substituted for the unknown quantity, reduce the first member to zero, or the proposed equation to the form of 0=0; because every equation may, designating the highest power of the unknown quantity by x", be exhibited under the form

xTM”+Axm−1+Bæт¬2+Сæm¬3+ . . . Tx+V=0. (1), A, B, C, . . . T, V, being known quantities. And the resolu tion of an equation is the method of finding all the roots, which will answer the required condition.

474. This being premised, it may now be shown, that if a be a root of the equation (1), the left hand member of that equation will be exactly divisible by x- -a.

For if a be substituted for x, agreeably to the above definition, we shall necessarily have

am+Aam-+Bam-3+Cam-3+... Ta+V=0.

And consequently, by transposition,

V=-am-Aam-1-Bam-2 — Cam—3—

-Ta.

Whence, if this expression be substituted for V in the first equation, we shall have, by uniting the corresponding terms, and placing them all in a line.

(xm-am)+A(xm—1—am-1)+B(xm—2—am-2)+T(x-a)=0. Where, since the difference of any two equal powers of two different quantities is divisible by the difference of their

roots (Art. 108), each of the quantities (xTM—am), (xm—1—am−1), (2m-2-am-2), &c. will be divisible by x-a. And, therefore, the whole compound expression

(x”—aTM)+A (xm— 1 — am—1)+B(xm—2—am—2)+ &c. =0, which is equivalent to the equation first proposed, is also divisible by X- -a; as was to be shown.

But if a be a quantity greater or less than the root, this conclusion will not take place; because, in that case, we shall not have

V=-am-Aam—1—Bam—2—Cam—3_

-Ta;

which is an equality obviously essential to the division in question.

475. The preceding proposition may be demonstrated," after the manner of D'ALEMBERT, as follows: In fact, designating by X, the polynomial, which forms the first member of the equation (1); then we shall always carry on the division of X by x-a, till we arrive at a remainder R, independent of x, since is only of the first degree in the divisor; so that, representing by Q the corresponding quotient, we shall have this identity,

X=Q(x-a)+R.

Now, by hypothesis, a substituted for a reduces the polynomial X to zero; and it is evident that the same substitution gives Q(x-a)=0; therefore we shall necessarily have 0=R: Hence x-a divides the equation (1), without a remainder.

Reciprocally, if the first member of any equation of the form X=0 be divisible by x—a, a is a root. In fact we have, according to this hypothesis, the identity X=Q(x-a), which, for x=a, gives X=0; therefore, (Art. 473), a is a root of the proposed equation.

COR. 1. Hence we may easily conclude, that if a be not a root of the equation (1), the first member will not be divisible by x-a.

COR. 2. And if the first member of the equation (1), be not divisible by -a, a is not a root of the proposed equation.

476. Supposing every equation to have one root, or value of the unknown quantity, it can then be shown, that any proposed equation will have as many roots as there are units in the index of its highest term, and no more. For let a, according to the assumption here mentioned, be a root of the equation (1),

xTM+Axm¬1+Bxm−2+Cxm¬3+ . . . +Tx+V=0.

Then, since by the last proposition this is divisible by x-a, it will necessarily be reduced, by actually performing the operation, to an equation of the next inferior degree, or one of the former

xm-1+A'xm-2+B'xm¬3+C'xm-4+ . . T'x+V'=0. And as this equation, by the same hypothesis, has also a root, which may be represented by a', it will likewise be reduced, when divided by x-a', to another equation one degree lower than the last; and so on.

Whence, as this process can be continued regularly in the same manner, till we arrive at a simple equation, which has only one root, it follows that the proposed equation will have

m roots

and that its successive divisors, or the factors of which it is composed, will be

x—a, x—a', x—a", x—a"",

z-a(m-1)',

being equal in number to the units contained in the index m of the highest terin of the equation.

COR. If the last term of an equation vanishes, as in the forin am+Am-'+Bæ12+...+Tr=0, it is evident that x=0 will satisfy the proposed equation; and consequently O is one of its roots. And if the two last terms vanish, or the equation be of the form "+Axm-'+B1⁄2·3¬2+ . . . +Sï3=0, two of its roots are 0; and so on. See, for another demonstration of the preceding proposition, Bonnycastle's Algebra,

vol. ii. 8vo.

477. Since it appears (Art. 474), that every equation, when all its terms are brought to one side, is exactly divisible by the unknown quantity in that equation minus either of its roots, and by no other simple factor, it is evident that the equation

-1

+Am-Bm-2+C2m-s+.. Tx+V=0. (1), of which a, b, c, d, . . . l, are supposed to be its several roots, is composed of as many factors,

(x—a) (x—b) (x—c) (x—d) .. (x—l) . (2),

as the equation has roots; and that it can have no other factor whatever of that form.

478. Whence, as these two expressions are, by hypothesis, identical, the proposed equation, by actually multiplying

the above factors, and arranging the terms according to the powers of x, will become

which form is general, whatever may be the different signs of the roots, or of the terms of the equation; taking a, b, c, &c. as well as A, B, C, &c. in + or - as they may happen to be.

479. Hence, since the two equations (1), (3), are identical, the coefficients of the like powers of x, are equal; and consequently, the following relations between the coefficients and roots will be sufficiently obvious.

I. The sum of all the roots of any equation, having its terms arranged according to the order of the powers of the unknown quantity, is equal to the coefficient of the second term of that equation, with its sign changed.

II. The sum of the products of all the roots, taken two and two, is equal to the coefficient of the third term, with its proper sign;

and so on.

III. The continued product of all the roots, is equal to the last term, taken with the same or a contrary sign, according as the equation is even or odd.

-

480. It is very proper to observe, that we cannot have all at once x=a, x=b, x=c, &c. for the roots of any equation as in the formula (2); except when a=b=c=d, &c., that is, when all the roots are equal. The factors - à, x − b, x—c, &c. exist in 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.

481. To this we may likewise add, that, if the roots of any equation be all positive, as in formula (2), where the factors are of the form

(x-a) (x—b) (x—c) (x−d).... (x-1)=0, the signs of the terms will be alternately + and -; as will readily appear from performing the operation required.