(Art. 194.) When the coefficient of the highest power is not unity, we may transform the equation into another, (Art. 166.), in which the coefficient of the first term is unity, and all the other coefficients whole numbers; but it is more direct and concise to modify the rule to suit the case. If the coefficient of the first power is c, the first divisor will be (cr+A)r+B, in place of (r+A)r+B. In place of (3r+s+A)s, to correct the first trial divisor, we must have (3cr+cs+A)s; and, in general, in place of using 3 times the root already found, we must use 3c times the root; and, in place of the square of any figure, as 12, s2, &c., we must use cr2, cs2, &c. EXAMPLES. 5. Find one root of the equation, 3x3+2x2+4x=75. By trial, we find that x must be more than 2, and less than 3; therefore r=2, c=3, A=2 B=4. N r stu • R is a symbol to represent the entire root, as far as determined. 6. Find one root of the equation, 5x-6x2+3x=-85. Ans. x=-2.16399-. 7. Find one root of the equation, 12x+x2-5x=330. Ans. x=3.036475+. s. Find one root of the equation, 5x3+9x2-7x=2200. Ans. x=7.1073536+. 9. Find one root of the equation, 5x3-3x2-2x=1560. Ans. x=7.0086719+. (Art. 195.) This principle of resolving cubic equations may be applied to the extraction of the cube root of numbers, and indeed gives one of the best practical rules yet known. For instance, we may require the cube root of 100. This gives rise to the equation x2+Ax2+Bx=100; in which A=0, and B=0, and the value of x is the root sought. As A and B are each equal to zero, the rule under (Art. 193.) may be thus modified. 1st. Keeping the symbols as in (Art. 193.), and finding r by trial, r2 will be the first divisor, and 3r2 is B', or the first TRIAL divisor. 2d. By means of the dividend (so called), and the first trial divisor, we decide s the next figure of the root. 3d. Then (3r+s)s; that is, three times the portion of the root already found, with the figure under trial annexed, and the sum multiplied by the figure under trial, will give a sum, which, if written two places to the right, under the last trial divisor, and added, will give the next complete divisor. 4th. After we have made use of any complete divisor, write the square of the last quotient figure under it; the sum of the three preceding columns is the next trial divisor; which use, and render complete, as above directed, and so continue as far as necessary.* In case of approximate roots after three or four divisors are found, we may find two or three more figures of the root, with accuracy, by simple division. 3. Extract the cube root of 1352605460594688. Ans. 110592. 4. Extract the cube root of 5382674. Ans. 175.25322796. 5. Extract the cube root of 15926.972504. Ans. 25.16002549 6. Extract the cube root of 91632508641. 7. Extract the cube root of 483249. Ans. 4508.33859058. (Art. 196.) The method of transforming an equation into another, whose roots shall be less by a given quantity, will resolve equations of any degree; and for all equations of higher degrees than the third, we had better use the original operation, as in (Art. 192.), and attempt no other modification than conceiving the absolute term to constitute the second member of the equation; and the difference of the numbers taken in the last column in place of their algebraic sum. The following operation will sufficiently explain: N. B. We went through the first and second transformations in full. Had we been exact, in the third, we should have added .08 to 39.2, and multiplied their sum, (39.28), by .08, giving 3.1424; we reserve 3. only to add to the next column. By a similar operation we obtain 46. to add to the next column. EXAMPLES. 1. Given x—x2—x3—x1-500—0, to find one value of x. Ans. 4.46041671 2. Given x1—5x3-+9x=2.8, to find one value of x. Ans. .32971055072 3. Given 20x+1122-9x3—x1—4, to find one value of x. Ans. .17968402502 4. Required the 5th root of 5000; or, in other terms, find one root of the equation 25-5000. Ans. 5.49280+ |