5 +28 +51 +32 -1 (-2 +18+15+ 2 -5 -5=1" R. Comparing this with the general equation (Art. 408,) we find A=5, B1=—12, C1=+3, D ̧=+4, and E1=-5. ... 5y-12y3+3y2+4y-5-0 is the transformed equation required. 3. Find the equation whose roots are less by 1.7 than those of the equation x3-2x2+3x-4=0. If we transform this equation into another whose roots are less by 1, the resulting equation is y3+y2+2y—2—0. We may then transform this into another whose roots are less by .7, and the result will be the equation required, or, the whole operation may be performed at once as follows: Hence, the required equation is y3+3.1y2+4.87y+.233 =0. It is generally preferable to perform the operation by successive transformations, using only one figure at a time, as there is less liability to error. It is not necessary, however, after each transformation to arrange the coëfficients in a horizontal line. Thus, the two transformations necessary in the preceding example may be combined as follows: 4. Find the equation whose roots are each less by 3 than the roots of x3-27x-36-0. Ans. y3+9y2-90=0. 5. Find the equation whose roots are each less by 3 than the roots of x-27x2-14x+120=0. Ans. y+12y+27y2-68y-84-0. 6. Required the equation whose roots are less by 5 than those of the equation x-18x3-32x2+17x+9=0. Ans. y1+2y3-152y2-1153y-2331=0. 7. Required the equation whose roots are less by 1.2 than those of the equation x-6x1+7.4x3+7.92x2 — 17.872x -.79232=0. Ans. y5—7y3+2y—8=0. Transform the following equations into others wanting the 2nd term. (See Art. 407.) Ans. y3+3y2—20—0, or y3—3y2—16=0. 13. x3-4x2+5x-2=0. Ans. y3—y2=0, or y3+y2-27=0. ART. 411, PROP. III.— To determine the law of Derived Polynomials. Let X represent the general equation of the nth degree; that is, If we substitute x+h for x, and put X, to represent the new value of X, we have X,=(x+h)"+A(x+h)n−1+B(x+h)n−2+, &c., and if we expand the different powers of x+h by the Binomial theorem, we have X,= +Axn¬1+(n−1)Axn−2 +(n−1)(n—2) Ax2¬3 +Bạn-2+(n-2)B-3 tìn−2(3) Ban +, &c. But the first vertical column is the same as the original equation, and if we put X', X", X"", &c., to represent the succeeding columns, we have X = x2+Ax2-1+Bxn−2+, &c., X'=nx11+(n−1)Ax”―2+(n—2)Bx"¬3+, &c., By substituting these in the development of X,, we have The expressions X', X", X"", &c., are called derived polynomials of X, or derived functions of X. X' is called the first derived polynomial of X, or first derived function of X; X" is called the second, X"" the third, and so on. It is easily seen that X' may be derived from X, by multiplying each term by the exponent of x in that term, and diminishing the exponent by unity. And each succeeding polynomial may be derived from that which precedes it by the same law. ART. 412. Cor. If we transpose X we have X,—X=X'h X" + h2+, &c. Now it is evident that h may be taken so small 1.2 X" 1.2 that the sign of the sum X'h+, h2+, &c., shall be the same as the sign of the first term X'h. For, since Xh+X"h2+, &c., =h(X'+}X"h+, &c.), if h be taken so small, that X′′h+}X'''h2+, &c., becomes less than X' (their magnitudes alone being considered), the sign of the sum of these two expressions must be the same as the sign of the greater X'. ART. 413. By comparing the transformed equation in Art. 406, with the development of X, in Art. 411, it is easily seen that X, may be considered the transformed equation, y corresponding to x, and r to h. Hence, the transformed equation may be obtained by substituting the values of X, X', &c., in the development of X,. As an example, let it be required to find the equation whose roots are less by 1 than those of the equation x3-7x+7=0. Observing that h=1, and substituting these values in the equation X,=X+X'h+X" n2+ X"" h3+, &c., we have 1.2 1.2.3 1 X ̧=(x3—7x+7)+(3x2—7)1+(6x); + 6 1.2 1 2.3' =x3-3x2—4x+1, in which the value of x is equal to that of x in the given equation diminished by 1. By this method the learner may solve the examples in Art. 410. ART. 414. To determine the equal roots of an equation. We have already seen (Art. 396, Rem. 2,) that an equation may have two or more of its roots equal to each other. Thus, the equation x3—6x2+12x-8=0, or (x—2)(x—2)(x—2) =(x-2)3=0, has three roots, each of which is 2. We now pro pose to determine when an equation has equal roots, and how to find them. If we take the equation (x—2)3=0 Its first derived polynomial is 3(x—2)2=0. (1) Hence, we see that if any equation contains the same factor taken three times, its first derived polynomial will contain the same factor taken twice; this last factor is, therefore, a common divisor of the given equation, and its first derived polynomial. In general, if we have an equation X=0, containing the factors (x-a)m(x-b)", its first derived polynomial will contain the factors m(x-a)m▬1n(x—b)n−1 ; that is, the greatest common divisor of the given equation, and its first derived polynomial will be (x—a)m—1 (x—b)n-1, and the given equation will have m roots, each equal to a, and n roots, each equal to b. Therefore, to determine whether an equation has equal roots, find the greatest common divisor between the equation and its first derived polynomial. If there is no common divisor the equation has no equal roots. If the greatest common divisor contains a factor of the form x-a, then it has two roots equal to a; if it contains a factor of the form (x-a)2 it has three roots equal to a, and so on. If it has a factor of the form (x—a)(x—b) it has two roots equal to a, and two equal to b; and so on. Ex. 1. Given the equation x3-x2-8x+12=0, to determine whether it has equal roots, and if so, to find them. We have for the first derived polynomial (Art. 411), 3x2-2x-8. The greatest common divisor of this and the given equation (Art. 108) is x-2. Therefore, the equation has two roots equal to 2. Now since the equation has two roots equal to 2, it must be divisible by (x-2)(x—2), or (x—2)2. (Art. 395.) Whence, x3-x2-8x+12=(x−2)2(x+3)=0, and x+3=0, or x=-3. It is evident that when an equation contains other roots besides the equal roots, that these may be found, and the degree of the equation depressed by division (Art. 395), after which the unequal roots may be found by other methods. |