296. Instead of making the second term disappear, it may be required to find an equation which shall be deprived of its third, fourth, or any other term. This is done, by making the co-efficient of u corresponding to that term equal to 0. For example, to make the third term disappear, we make, in the above-transformed equation from which we obtain two values for x, which substituted in the transformed equation reduce it to the form um + P'um-1 + R'um-3... + T'u + U' = 0. Beyond the third term it will be necessary to resolve an equation of a degree superior to the second, to obtain the value of x'; and to cause the last term to disappear, it will be necessary to resolve the equation x'm + Px/m-1... + Tx + U = 0, which is what the given equation becomes when a' is substituted for x. It may happen that the value P m which makes the second term disappear, causes also the disappearance of the third or some other term. For example, in order that the third term may disappear at the same time with the second, it is necessary that the value of a which results from the equation Now, if in this last equation, we replace x by m-1 P2 m 2 P2 m -(m-1)+Q = 0, or (m - 1) P2 - 2mQ= 0; the disappearance of the second term will also involve that of the the third. Formation of derived Polynomials. 297. The relation x = u + x', which has been used in the two preceding articles, indicates that the roots of the transformed equations are equal to those of the given equation, increased or diminished by a certain quantity. Sometimes this quantity is introduced into the calculus, as an indeterminate quantity, the value of which is afterward determined by requiring it to satisfy a given condition; sometimes it is a particular number, of a given value, which expresses a constant difference between the roots of a primitive equation and those of another equation which we wish to form. In short, the transformation, which consists in substituting u + x' for x, in a given equation, is of very frequent use in the theory of equations. There is a very simple method of obtaining, in practice, the transformation which results from this substitution. To show this, let us substitute for x, u + x' in the equation xm + Pxm-1 + Qxm-2 + Rxm-3 + ... Tx + U = 0; then, by developing, and arranging the terms according to the ascending powers of u, we have If we observe how the co-efficients of the different powers of u are composed, we shall see that the co-efficient of uo, is what the first member of the given equation, becomes when a' is substituted in place of x; we shall denote this expression by X'. The co-efficient of u1 is formed from the preceding term X', by multiplying each term of X by the exponent of x in that term, and then diminishing this exponent by unity; we shall denote this co-efficient by Y'. The co-efficient of u2 is formed from Y', by multiplying each term of Y' by the exponent of x in that term, dividing the product by 2, and then diminishing each exponent by unity. Repre senting this co-efficient by Z' , we see that Z' is formed from Y', in the same manner that Y' is formed from X'. In general, the co-efficient of any power of u, in the abovetransformed equation, may be found from the preceding co-efficient in the following manner: viz., By taking each term of that co-efficient in succession, multiplying it by the exponent of x', dividing by the number which marks the place of the co-efficient, and diminishing the exponent of x' by unity. The law by which the co-efficient are derived from each other, is evidently the same as that which governs the formation of the terms of the binomial formula (Art. 203). The expressions, Y', Z', V', W..... are called derived polynomials of X', because each is derived from the one which precedes it, by the same law as that by which Y' is deduced from X. Hence, generally, A derived polynomial is one which is deduced from a given polynomial, according to a fixed and known law. Recollect that X is what the given polynomial becomes when x is substituted for x. Y' is called the first-derived polynomial; Z' is called the second-derived polynomial; V' is called the third-derived polynomial. We should also remember if we make u = 0, we shall have, x = x, whence X' will become the given polynomial, from which the derived polynomials will then be obtained. 298. Let us now apply the above principles in the following EXAMPLES. 1. Let it be required to find the derived polynomials from the equation 3x4 + 6x3 3x2 + 2x + 1 = 0 = X. Now, u being zero, and x = x, we have from the law of forming the derived polynomials, X = X = 3x4+ 6x3-3x2+2x+1; Y' = 12x3 + 18x2-6x+2; Z'=36x2+36x V' = 72x + 36; W' = 72. 6; It should be remarked that the exponent of x in the terms 1, 2 6, 36, and 72, is equal to 0; hence, each of those terms dis. appears in the following derived polynomial. 2. Let it be required to cause the second term to disappear in the equation and the operation is reduced to finding the values of the co-efficients Now, it follows from the preceding law for derived polynomials, that (3)4-12.(3)3+17.(3)2-9. (3)1+7, or X' =−110; X' into another equation, the roots of which shall exceed those of the given equation by unity. Make, x = -1; whence x = -1; and the transformed equation will be of the form X' = 4.(-1)3- 5. (-1)2 +7.(-1)1-9, or X' = -25; 4. What is the transformed equation, if the second term be made to disappear in the equation 05 10x4 + 7x3+4x-90? Ans. u5 118u2 3343 152-73 = 0. 5. What is the transformed equation, if the second term be made to disappear in the equation into another, the roots of which shall be less than the roots of the |