627. It sometimes happens that the division of the last term of the first transformed equation by the coefficient of in that equation gives a quotient greater than unity. In that case, as where the signs of these terms are alike, we obtain another figure of the root by the method used to obtain the integral part of the root.
628. If in any transformed equation after the first the signs of the last two terms are the same, the figure of the root used in making the transformation is too large and must be diminished until these terms have unlike signs.
629. If in any transformed equation the coefficient of the first power of the unknown quantity is zero, we may obtain the next figure of the root by using the coefficient of the second power of the unknown quantity as a divisor and taking the square root of the result.
630. Negative incommensurable roots may be found by transforming the equation into one whose roots shall he positive [Art. 580], and finding the corresponding root. This result with its sign changed will be the root required.
631. Any Root of Any Number. By Horner’s Method we can find approximately any root of any number; for placing n a equal to x we have for solution the equation x^n = a, or x^n-a=0.
EXAMPLES XLVIII. 1.
Compute the root which is situated between the given limits in the following equations :
1. x^3 + 10x^2+ 6x -120=0; root between 2 and 3. 2. x^3-2x-5=0; root between 2 and 3. 3. x^4 - 2x^3 + 21x -23=0; root between 1 and 2. 4. x^3 +x - 1000 = 0; root between 9 and 10. 5. x^3+ x^2+x-100 =0; root between 4 and 5. 6. 2x^3 +3x^2 -4x-10=0; root between 1 and 2. 7. x^3 - 46 x^2 - 36x+18=0; root between 0 and 1.
