ALGEBRA. [CH. XL the second line of the work above are, in order, the coefficients of the depressed equation with signs changed. We thus have the following method: Divide the last term by r and add the quotient to the coefficient of x. Divide this sum by r and add the quotient to the coefficient of x. Divide this sum by r and add the quotient to the coefficient of x, and so on. The last quotient will be — bə If at any stage the division be not exact, this fact at once shows that the number being tested is not a root. Ex. Solve the equation 2 - 4 x2 − 9 x + 36 = 0. Substituting 1 and -1, we find that they are not roots. Testing2 by Newton's Method, 1 -4 -9 36(-2 - 27 36 27 is not divisible by 2, 2 is not a root. Testing 3, 1 4 9 36(3 12 3 3 36 Since the division at each step is exact and the last quotient is - 1, 3 is a root, and the depressed equation is x2-x-12= 0. The roots of this equation are of the given equation are 3, 3, 4. 3, 4. Therefore the roots EXERCISES VI. Find a real root of each of the following equations, and solve the depressed equation : We have f (x + h) = αo(x + h)" + a1(x + h)n−1 + A2(x + h)n-2 -2 + ··· + An-1(x + h) + ɑn + An-12 + an +h[na,xn−1 + (n − 1) α1x2 -2 + (n − 2) α2x2-3 + + an-1] The coefficient of h is derived from ƒ (x) as follows: Multiply each term of f (x) by the exponent of x in that term and sub tract 1 from the exponent. Thus, from aox" we derive nax-1; from a12-1 we derive (n−1) a1x-2; and so on. On this account the coefficient of h is called the First Derived Function,. and is represented by ƒ'(x). Observe that from an-12 we derive 1 x an-12o, = An-1. These successive coefficients are therefore called the Second Derived Function, the Third Derived Function, and so on; and are represented by ƒ"(x), ƒ'''(x), and so on, respectively. Notice that f'(x) is of degree one lower than ƒ(x); that ƒ"(x) is of degree two lower than ƒ (x); and so on. h2 We now have f(x + h) = f(x) + hf'(x) + · f'(x) + 12 In like manner ƒ (x + h) could have been expanded in terms of ascending powers of x. Ex. Given f(x) = x3 − 2 x2 + x + 3, find ƒ (x + 2). In the terms of the second member, 2 is to be substituted for x in f(x), f'(x), and so on. We now have ƒ (x) = x3 − 2 x2 + x + 3, ƒ' (x) = 3 x2 − 4 x + 1, f" (x)=6x-4, ƒ""'(x) = 6, fiv(x) = 0, etc. Whence ƒ(2) = 5, ƒ'(2) = 5, ƒ"(2) = 8, ƒ"''(2) = 6. f(x+2) = 5 + 5 x + 4 x2 + x3. Multiple Roots. 33. If r be k times a root of f(x) = 0, then x − r is k times a factor of f(x), or (x − r)* is a factor. wherein F(x) stands for the product of the remaining factors of ƒ (x), and does not contain the factor x — r. We now have f(x + h) = (x − r + h) * F (x + h) Expanding ƒ(x + h) and F(x + h) by the preceding article, and . (x − r + h)* by the binomial theorem, we have = (x − r)1F(x) + h[k(x − r)k -1F(x) + (x − r)*F'(x)] + Equating coefficients of h, f'(x)= k(x − r)* −1 F(x) + (x − r)*F′(x) = (x − r)*−1[kF(x) + (x − r) F' (x)]. We now see that f'(x) contains zr as a factor k-1 times; therefore f'(x) = 0 has k 1 roots equal to r. That is, if f(x) = 0 have a multiple root, then f'(x) = 0 has the same root repeated one less time. 34. It follows also from the preceding article that (x − r)*-1 is the H.C.F. of f(x) and f'(x). We can therefore find the multiple roots of any given equation, ƒ(x) = 0, by finding the roots of the H. C. F. of ƒ (x) and f'(x) equated to 0. If f(x) and f'(x) have no common factor, f(x) = 0 does not have a multiple root. Ex. The equation x1 — 9 x3 + 23 x2 – 3 x − 36 = 0 has multiple roots. Solve the equation. We have and ƒ (x) = x1 − 9 x3 + 23 x2 − 3 x — 36, f'(x) = 4x3- 27 x2 + 46 x 3. The H.C.F. of ƒ (x) and ƒ'(x) is found to be x - 3. Therefore (x-3)2 is a factor of f(x), and 3 is a root twice of the given equation. Removing the root 3 twice, Solve the following equations, which have multiple roots: Graphic Representation. 35. Two perpendicular straight lines divide the plane in which they lie into four parts, called Quadrants. Thus, the lines XX, YY', in Fig. 5, divide the plane of the paper into four quadrants: B. FIG. 5. XOY called the first quadrant; The position of a point in a plane is known if its distance from these two fixed lines, and the quadrant in which it lies, be known. Thus, the position of a point P, in the first quadrant is known, if we know the distances AP, and B1P1; that of the point P2 in the second quadrant, if we know the distances AP, and B.P2. 2 36. The two lines X'OX and Y'OY are called Axes of Reference, and the point O is called the Origin. The distance of a point P from the axis Y'OY, measured along or parallel to the axis X'OX, is usually designated by the letter x, and is called the Abscissa, or the x of the point. Thus, the x of the point P, is OA1, = B1P1. The distance of a point P from the axis X'OX, measured along or parallel to the axis Y'OY, is usually designated by the letter y, and is called the Ordinate, or the y of the point. Thus, the y of the point P1 is OB1, = ¡P1. The abscissa and the ordinate of the point are together called the Coördinates of the point. The axis X'OX is called the Axis of Abscissas, or the x-axis. The axis Y'OY is called the Axis of Ordinates, or the y-axis. 37. The quadrant in which a point lies is determined as follows: |