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to; and the remaining
roots are negative. The rule alfo may be demonftrated.
Note. The impoffible roots in this rule are fuppofed to be either pofitive or negative.
In this example of a numeral equation x4—10x +35x2-50x+24=0, the roots are, +1, +2, +3, +4, and the preceding obfervations with regard to the figns and coefficients take place.
Cor. If a term of an equation is wanting, the positive and negative parts of its coefficient must then be equal. If there is no abfolute term, then fome of the roots must beo, and the equation may be depreffed by dividing all the terms by the lowest power of the unknown quantity in any them. In this cafe alfo, x-o=0, x―0=0, &c. be confidered as fo many of the
component simple equations, by which the given equation being divided, it will be depreffed fo many degrees.
CHA P. II.
Of the Transformation of Equations.
'HERE are certain transformations of
equations neceffary towards their folution; and the most useful are contained in the following propofitions.
The affirmative roots of an equation become negative, and the negative become affirmative, by changing the signs of the alternate terms, beginning with the fecond.
Thus the roots of the equation x-x3 19x2+49x-30=0 are +1, +2, +3,—5, whereas the roots of the equation, x++x319x2—49x—30=0,are—1,—2,—3, +5.
The reason of this is derived from the compofition of the coefficients of these terms, which confift of combinations of odd numbers of the roots, as explained in the preceding chapter.
Prop. II. An equation may be tranfformed into another that fhall have its roots greater or less than the roots of the given equation by fome given difference.
Let x be the unknown quantity of the equation, and e the given difference; let y=xe, then x=ye; and if for x and
its powers in the given equation, ye and its powers be inferted, a new equation will arife, in which the unknown quantity is y, and its value will be xe; that is, its roots will differ from the roots of the given equation by e.
Let the equation propofed be x—px2+ qx-r=0, of which the roots must be diminished by e. By inferting for x and its powers, ye and its powers, the equation required is,
Cor. I. From this transformation, the fecond, or any other intermediate term, may be taken away; granting the refolution of equations.
Since the coefficients of all the terms of the transformed equation, except the first, involve the powers of e and known quantities only, by putting the coefficient of any term equal to o, and refolving that equation, a value of e may be determined; which being fubftituted, will make that term to vanish.
Thus, in this example, to take away the fecond term, let its coefficient, 3e-p=0,and e=p, which being fubftituted for e, the new equation will want the second term. And univerfally, the coefficient of the first term of a cubic equation being 1, and x being the unknown quantity, the second term may be taken away, by supposing x=y=p, p being the coefficient of that
Cor. 2. The fecond term may be taken away by the folution of a fimple equation, the third by the folution of a quadratic, and fo on.
Cor. 3. If the fecond term of a quadratic equation be taken away, it will become a pure equation, and thus a folution of quadratics will be obtained, which coincides with the folution already given in Part I.
Cor. 4. The last term of the transformed equation is the fame with the given equation, only having e in place of x.
Prop. III. In like manner may an equation be transformed into another, of which the roots shall be equal to the roots of the given equation, multiplied or divided by a given quantity.
Let x be the unknown letter in the given equation, and y that of the equation wanted; alfo let e be the given quantity. To multiply the roots, let xe=y, and
To divide the roots let =y,andx=ye.