we have x2+x-2, which being placed equal to zero, gives the two roots x = 1, x = −2, or the two factors, x Hence we have. x + 2. 1 and x2 +3x3 + x2-3x-2=(x + 1)2 (x − 1)(x+2). Therefore, the first member of the proposed equation is equal to (x + 1)3 (x − 1)2 (x + 2)2 ; that is, the proposed equation has three roots equal to 1, two equal to +1, and two equal to 2. 4. What are the equal factors of the equation - = 0. ·7x6+10x5 + 22x2 - 43x3 35x2+48x + 36 5. What are the equal factors in the equation 303. To eliminate between two equations of any degree whatever, involving two unknown quantities, is to obtain, by a series of operations, performed on these equations, a single equation which contains but one of the unknown quantities, and which gives all the values of this unknown quantity that will, taken in connexion with the corresponding values of the other unknown quantity, satisfy at the same time both the given equations. This new equation, which is a function of one of the unknown quantities, is called the final equation, and the values of the unknown quantity found from it, are called compatible values. Elimination by Means of Indeterminate Multipliers. 304. Let there be the equations ax + by — c = 0, a'x + b'y-c' = 0. If we multiply the first by m, and subtract the second from the product, we have (ma — a') x + (mb — b′) y — mc + c = 0 . . . (1). Now, since the value of m is entirely arbitrary, we may give it such a value as to render the co-efficient of x zero, which gives Substituting in equation (2) the value of m, and we have Had we chosen to attribute to m such a value as to render the co-efficient of y zero in equation (1), we should have had Substituting in equation (3) the value of m, we obtain The above values for x and y are the same as those determined in Art. 97. The principle explained above is applicable to three or more equations, involving a like number of unknown quantities. 305. Of all the known methods of elimination, however, the method of the common divisor is, in general, the best; it is this method which we are going to develop. Let f(x, y) =0 A, and f'(x, y) = 0 = B, be any two equations whatever, in which fand f' denote any functions of x and y. Suppose the final equation involving y obtained, and let us try to discover some property of the roots of this equation, which may serve to determine it. be one of the values of y which will satisfy both the given equations. This is called a compatible value of y. It is plain, that, since this value of y, in connexion with a certain value of x, will satisfy both equations, that if it be substituted in them, there will result two equations involving x alone, which will admit of at least one common value of x; and to this common value there will correspond a common divisor involving x (Art. 279). This common divisor will be of the first, or of a higher degree with respect to x, according as the particular value of y = a corresponds to one or more values of x. Reciprocally, every value of y which, substituted in the two equations, gives a common divisor involving x, is necessarily a compatible value, because it then evidently satisfies the two equations at the same time with the value or values of x found from this common divisor when put equal to 0. 306. We will remark, that, before the substitution, the first members of the equations cannot, in general, have a common divisor which is a function of one or both of the unknown quantities. For, let us suppose for a moment that the equations Making separately D = 0, we obtain a single equation involving two unknown quantities, which can be satisfied with an infinite number of systems of values. Moreover, every system which renders D equal to 0, would at the same time cause A'D, B'D to vanish, and would consequently satisfy the equations A=0 and B = 0. Thus, the hypothesis of a common divisor of the two polynomials A and B, containing x and y, would bring with it as a consequence that the proposed equations were indeterminate. Therefore, if there exists a common divisor, involving x and y, of the two polynomials A and B, the proposed equations will be indeterminate, that is, they may be satisfied by an infinite number of systems of values of x and y. Then there would be no data to determine a final equation in y, since the number of values of is infinite. If the two polynomials A and B were of the form y D being a function of x only, we might conceive the equation D= 0 resolved with reference to x, which would give one or more values for this unknown. in the equations Each of these values substituted A' x D = 0 and B' x D = 0, would verify them, without regard to the value of y, since D must be nothing, in consequence of the substitution of the value of x. Therefore, in this case, the proposed equations would admit of a finite number of values for x, but of an infinite number of values for y, and then there could not exist a final equation in y. Hence, when the equations A = 0, B = 0, are determinate, that is, when they admit only of a limited number of systems of values for x and y, their first members cannot have for a common divisor a function of these unknown quantities, unless a particular substitution has been made for one of these quantities. 307. From this it is easy to deduce a process for obtaining the final equation involving y. Since the characteristic property of every compatible value of y is, that being substituted in the first members of the two equations, it gives them a common divisor involving x, which they had not before, it follows, that if to the two proposed polynomials, arranged with reference to x, we, apply the process for finding the greatest common divisor, we shall generally not find one. But, by continuing the operation properly, we shall arrive at a remainder independent of x, but which is a function of y, and which, placed equal to 0, will give the required final equation. For, every value y found from this equation, reduces to nothing the last remainder of the operation for finding the common divisor; it is, then, such, that substituted in the preceding remainder, it will render this remainder a common divisor of the first members A and B. Therefore, each of the roots of the equation thus formed, is a compatible value of y. of 308. Admitting that the final equation may be completely resolved, which would give all the compatible values, it would afterward be necessary to obtain the corresponding values of x. Now, it is evident that it would be sufficient for this, to substitute the different values of y in the remainder preceding the last, put the polynomial involving a which results from it, equal to 0, and find from it the values of x; for these polynomials are nothing more than the divisors involving x, which become common to A and B. But as the final equation is generally of a degree superior to the second, we cannot here explain the methods of finding the values of y. Indeed, our design was principally to show that, two equations of any degree being given, we can, without supposing the resolution of any equation, arrive at another equation, containing only one of the unknown quantities which enter into the proposed equation. If it were required to find the final equation in x, we observe that x and y enter in the same manner into the original equations; hence, may be changed into y and y into x, without destroying the equality of the members. Therefore, 4x6 6x4+3x2 -1=0 is the final equation in x. |