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, or the whole operation may be performed at once, as follows: Hence, the equation is y3+3.1y2+4.87y+.233-0. 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. Required the equation whose roots are less by 5 than those of the equation x-18x3-32x2+17x+9=0. Ans. y+2y3—152y2-1153y-2331=0. 6. Required the equation whose roots are less by 1.2 than those of the equation x-6x+7.4x3+7.92x2-17.872 -.79232-0. Ans. y5-7y3+2y—8—0. Transform the following equations into others wanting the 2d term. (See Art. 407.) 7. x3-6x2+7x-2=0. 8. x3-6x2+12x+19=0. Ans. 3-5y-4=0. Ans. y3+27=0. Transform the following equations into others wanting the 3d term: 9. x3-6x2+9x-20-0. Ans. y3+3y2-20-0, or y3-3y2-16-0. 10. x3-4x2+5x-2=0. Ans. y-y=0, or y3+y2-4=0. 411. Proposition 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 X1 to represent the new value of X, we have X1=(x+h)”+A(x+h)n−1+B(x+h)n-2+, etc., and if we expand the different powers of x+h by the binomial theorem, we have X1= But the first vertical column is the same as the original equation, and if we put X', X", X", etc., to represent the succeeding columns, we have X = x2+Ax2-1+Bxn-2+, etc., Xnxn-1+ (n-1) Ax"-2(n-2)B"-3, etc., By substituting these in the development of X1, we have The expressions X', X", X", etc., 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, X" from X', etc., by multiplying each term by the exponent of x in that term, and diminishing the exponent by unity. 412. Corollary.-If we transpose X, we have X-X X" =X'h+gh2+, etc. Now, it is evident that h may be 1.2 X" taken so small that the sign of the sum X'h+12h2+, etc., will be the same as the sign of the first term X'h. For, since X'h+X"h2+, etc., =h(X'+X"h+, etc.), if h be taken so small, that X′′h+1X"" h2+, etc., 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'. 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 tranformed equation may be obtained by substituting the values of X, X', etc., in the development of X1. As an example, 1 Let it be required to find the equation whose roots are less by 1 than those of the equation x3-7x+7=0. Here, X =x3-7x+7, X"=6x, X"=6, Xiv=0. Observing that h=1, and substituting these values in the equa the value of x is equal to that of x in the given equation diminished by 1. By this method, solve the examples in Art. 410. EQUAL ROOTS. 414. To determine the equal roots of an equation. We have already seen (Art. 396, Rem.) that an equation may have two or more of its roots equal to each other. We now propose to determine when an equation has equal roots, and how to find them. If we take the equation (x-2)3=0 (1), its first derived polynomial is 3(x-2)2=0. 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 equatfon, and its first derived polynomial. In general, if we have an equation X=0, containing the factors (x—a)TM (x—b)", its first derived polynomial will contain the factors m(x—a)m-1n(x—b)"−1; that is, the greatest common divisor of the given equation, and its first derived polynomial, will be (x—a)m—1 (x—b)"-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 G.C.D. 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) 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. 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 G.C.D. of this and the given equation (Art. 108) is x-2. Hence, x-2=0, and 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. Hence, when an equation contains other roots besides the equal roots, the degree of the equation may be depressed by division, and the unequal roots found by other methods. The following equations have equal roots; find all the roots. 5. x1-7x3+9x2+27x-54-0. Ans. x=3, 3, 3, −2. 6. x1+2x3-3x2-4x+4=0. Ans. -2, -2, +1, +1. 7. x1—12x3+50x2-84x+49=0. A. 3±√/2, 3±√2. 8. x3-2x+3x3-7x2+8x-3=0. Ans. 1, 1, 1, −±√—11. 9. x+3x5—6x1—6x3+9x2+3x-4=0. Ans. 1, 1, 1, −1, −1, −4. SUGGESTION.-In the solution of equations of high degree, the principles above explained may be extended. Thus, in the last example, the G.C.D. is x3—x2—x+1. Proceeding, we may, 1st, find the common measure of this and its first derived polynomial, and thus resolve into factors; or, 2d, find the G.C.D. of the first and second derived polynomials. If it is of the form x-a, one of the factors of the original equation will evidently be (x-a)3, etc. By the 1st method, we find x3—x2—x+1=(x−1)(x2—1)—(x—1)2 (x+1); by the 2d, (x-1)3 is a factor of the original equation; hence, (x-1)2 is a factor of x3-x2—x+1. LIMITS OF THE ROOTS OF EQUATIONS. 415. Limits to a Root of an Equation are any two numbers between which that root lies. A Superior Limit to the positive roots is a number numerically greater than the greatest positive root. |