is-a-b-c; in the biquadratic, whose roots are+a,+b,+c, and+d, it is-a-b-c-d, &c. 25. The coefficient of the third term in each, is the sum of all the products that can possibly arise by combining the roots, with their proper signs, two and two. Thus, in the cubic, the coefficient of the third term is+ab+ ac+bc; in the biquadratic, it is+ab+ac+ad+bc+bd+cd, &c. 26. The coefficient of the fourth term in each, is the sum of all the products that can possibly arise by combining the roots, with their signs changed, three by three. Thus, in the biquadratic, the coefficient of the fourth term is-abc-abd-acd-bcd. In like manner, in higher equations, the coefficient of the fifth term will be the sum of all the products of the roots, having their proper signs, combined four by four; that of the sixth term, the roots, with their signs changed, five by five, &c. 27. The last, or absolute term, is always the continued product of all the roots, having their signs changed. Thus, in the quadratic, whose roots are+a and+b, the last term is+ab (or—a×—b); in the cubic, the absolute term is — abc (=-ax-bx-c); in the biquadratic, the absolute term is + abcd (ax-bx-cx-d), &c. 28. The first term is always positive, and some pure power of x. 28. B. The second term is some power of x multiplied into -a,-b,-c, &c. and since x is affirmative, and each of these quantities negative, it follows that the second term itself is negative, since+x-produces —. 29. The third term will be positive, for its coefficient being the sum of the products of every two of the negative quantities -a,b,c, &c. and (since-x-produces +) therefore these sums, multiplied by any power of x, (which is always positive,) will always give a positive result. 30. For like reasons the fourth term will be negative, the fifth positive, the sixth negative, and so on; that is, when the roots are all positive, the signs of the terms of the equation will be alternately positive and negative and conversely, when the signs of the terms of the equation are alternately + and all the roots will be positive. : COR. Hence, if the signs of the even terms be changed, the signs of all the roots of the equation will be changed. 31. Let now the roots of the equations, above referred to, be supposed negative; that is, x=—a, x=—b, x=—c, x=—d, &c. then by transposition, x+a=o, x+b=o, x+c=o, x+d=o, &c. the product of these, or x+a.x+b.x+c.x+d, &c. will be an equation, having all its terms affirmative; for since all the quantities composing the factors are +, it is plain that the products will all be +. COR. Hence, when the signs of all the roots (in the above simple equations, having both terms on one side) are —, the signs of all the terms of the equation compounded of them will be+; and conversely, when the signs of all the terms of an equation are +, the signs of all its roots will be 32. If equations similar to the foregoing be generated, having some of the roots+, others it will appear, that there will be as many changes in the signs of the terms, (from+to−, or from to+,) as the equation has positive roots; and as many continuations of the same sign, (+and+, or — and —,) as the equation has negative roots: and conversely, the equation will have as many affirmative roots as it has changes of signs, and as mány negative roots as it has continuations of the same sign ". COR. It follows from what has been said, that every equation has as many roots as its unknown quantity has dimensions. To be particular; a quadratic has two roots, which are either both affirmative, both negative, or one affirmative and one This supposes the roots to be all possible. Every equation will have either an even number of impossible roots, or none: hence a quadratic will have both its roots possible, or both impossible; a cubic one or three possible roots, and two or none impossible; a biquadratic will have either four, two, or none of its roots possible, and none, two, or four, impossible; and the like of higher equations. An impossible root may be considered either as affirmative or negative. The difficulties attending the doctrine of impossible or imaginary roots, have hitherto bid defiance to the skill and address of the learned: a great number of theories and investigations have appeared, it is true; but our knowledge of the origin, nature, properties, &c. of imaginary roots is still very imperfect. The following Authors, among others, have treated on the subject, viz. Cardan, Bombelli, Albert Girard, Wallis, Newton, Maclaurin, James Bernoulli, Emerson, Euler, D'Alembert, Waring, Hutton, Sterling, Playfair, &c. negative. A cubic has three roots, which are either all affirmative, all negative; two affirmative, and one negative; or one affirmative, and two negative: and the like of higher equations. 33. If one root of an equation be given, the equation may be depressed one dimension lower; if two roots be given, it may be depressed two dimensions lower, and so on, by the following rule. RULE. When one root is given, transpose all the terms to one side, whereby the whole will=0; transpose in like manner the value of the root, then divide the former expression by the latter, and a new equation will arise=0, of one dimension lower than the given equation. EXAMPLES.-1. Let x3-9x+26x-24-o be an equation, whereof one of the roots is known; namely, x=3. By transposition x-3=0, divide the given equation by this quantity. Thus, x-3)x-9 x2+26x-24(x2-6x+8=0, the resulting equation, which being re −6 x2+26 x -6 x2+18 x solved by the known rule for quadratics, its two remain 8x-24 ing roots will be found, viz. 8x-24 x=4, and x=2. 2. Let x2+4x3 +19 x2— 160x=1400, whereof one root= 5, be given, to depress the equation. - Here by transposition, x+4x3 +19 x2-160 x-1400=0, and x+5=0; then, dividing the former by the latter, we have x+4x3+19 x2- 160 x 1400 x+5 sulting equation. · = x3 — x2 +24 x-280=0, the re 3. Given x=3 in the equation x2-5x+6=o, to depress it. Ans. x-2=0. 4. If x-4=o be a divisor of the equation x3- 4 x2-x+4=0, to depress the equation, and determine its two remaining roots. Ans. the resulting equation is x2-1=o, and its roots +1 and -1. When the absolute term of an equation=o, it is plain that one of the roots is o, and consequently the equation may be divided by the unknown quantity, and reduced one dimension lower. In like manner, if the two last terms be wanting, the equation may be reduced two dimensions lower; if three, three dimensions, &c. 5. To depress the equations x3-5x2+2x+8=0, and x23x2+18x+40=0, one root of the former being +4, and one of the latter -5. 34. If two of the roots be given, x+r=o, and x+s=o, the given equation being divided by the product of these, x+r.x+s, will be depressed thereby two dimensions lower; thus, 6. To depress the equation a3-5 x2+2x+8=0, two of its roots, -1 and +2, being given. Thus, x+1=0, and x-2=0; then x+1.x-2=x2—x—2, the divisor; wherefore is the resulting equation. 2 7. Given x3-3x2-46 x-72=0, having likewise two values of x, viz. -2 and -4, given, to depress the equation. Answer, x-9=0. 8. Given x4-4 x3 — 19 x2 + 46x+120=0, two roots of which - 35. To transform an equation into another, the roots of which uill be greater, by some given quantity, than the roots of the proposed equation. RULE I. Connect the given quantity with any letter, different from that denoting the unknown quantity in the proposed equation, by the sign and it will form a residual. II. Substitute this residual and its powers, for the unknown quantity and its powers in the proposed equation, and the result will be a new equation, having its roots greater, by the given quantity, than those of the equation given". a The truth of this rule is clear from the first example, where since y-3=x, it is plain that y=x+3, or that the equation arising from the substitution of y-3 for x will have its roots (or the values of y) greater by 3, than the values of x in the proposed equation: this will be still more evident, if both the given and the resulting equation be solved; the roots of the former will be found to be -7 and +3, those of the latter -4 and +6. Let it not be thought strange that the negative quantity -7, by being increased by 3, becomes -4, or a less quantity than it was before; for a negative quantity is said to be increased, in proportion as it approaches towards an affirmative value; thus, -3 is said to be greater than -4, -2 than -3, -1 than —2, and 0 than -1: in the present instance, it is plain that -7 added to 43 will give -4 for the sum. Hence, if the roots of an equation be increased by a quantity greater than the EXAMPLES.-1. Given x2+4x-21=0, to transform it into another equation, the roots of which are greater by 3 than those of the given equation. 2. Given the equation x3+x2-10x+4=0, to transform it into another, the roots of which are greater by 4 than the va + y2 8y+16 + x2=(y-412)... -10x=(y-4.-10=).... -10y+40 +8='. x3+x2-10x+8= + 8 y3-11 y2+30 y=0 This transformed equation is evidently divisible by y (or y+o, or y-o); therefore o is one of its roots: by this division it becomes y2—11 y+30=0, the two roots of which are +6 and +5; hence the three roots of the equation y3 —11 y2+30 y=o, being o +6, and +5, those of the proposed equation x3 + x2-10x+8=0 are known; for (since x=-y-4) its roots will be o—4, 6—4, and 5-4; or -4, +2, and +1. COR. Hence, when the roots of an equation are increased by a quantity equal to one of the negative roots, that root is taken away, or becomes o in the transformed equation; and in this case, the transformed equation may be depressed one dimension lower. 3. To increase the roots of the equation x3-6 x2 + 12 x−8 =o, by 1. greatest negative root, the negative roots will be changed into affirmative ones. It may be likewise useful to remark, that a deficient equation may be made complete by this rule. |