125. When the roots of an equation are integral, they may sometimes, be found with great facility, by seeking the divisors of the last term, and substituting them in place of the unknown quantity, till one or more be found which answer the conditions of the equation. (See art. 122.) When one root has been found, the equation may be depressed, by connecting that root, with its sign changed with the unknown quantity, and dividing the given equation by the sum. EXAMPLES. 1. Given x3-3x2+5x-15=0, to find the value of x. (Art. 119,) the roots if possible, are all positive.* Also, the divisors of 15 are 1, 3, 5, 15. Now, by substituting these for x, .. the roots are 3, +/-5, and —✅✔✅—5. 2. Given x3-2x2-5x+6=0, to find x: Results, 1, 3, -2. * The rules, in the foregoing articles, for determining the signs of the roots from the changes in the signs of the terms composing the equation, being founded on the supposition, that each root has but one sign, do not apply to impossible roots; because the negative signs under the radicals, when developed by multiplication, are combined with those of the roots, and therefore, change the signs of the terms of which the equation is composed. 3. Given 2+6x2-7x-60-0, to find x. Results, 3,-4,-5. 4. Given 3+3x-6x-8=0, to find x. 5. Given x3-2x+4=0, to find x. Results, 2,-1,-4. Results, 2, 1+√-1, 1-√-1. 6. Given 2-10x+35x-50x+24=0, to find x. Results, 1, 2, 3, 4. 7. Given x+-8x3+23x2—64x+120=0, to find x. Results, 5, 5, 2√√—2, −2√√✓—2. 126. When the roots are not integral, they may generally be determined by approximation.* For this purpose, various rules have been investigated. Among these the following is probably the most convenient in practice. The demonstration is given in the subsequent article. To find the root of a general equation. 1. If all the terms of the equation are not on one side, by transposition, place them so; and arrange them according to the powers of the unknown quantity, placing the highest power on the left hand. If any of the lower powers are not contained in the equation, consider each one omitted, as having a cipher for its co-efficient. 2. Place the co-efficients and the absolute number with their proper signs, in order, in a horizontal line. * If all the roots of the equation are impossible, this method is not applicable, (see art. 106,) but possible roots may always be approximated. N 3. Find by trial the first root figure, attending to its value as being units, tens, tenths, or hundreths, &c. and place it to the right of the absolute number. 4. Multiply the first co-efficient by the root figure, and add the product to the second co-efficient; multiply the sum by the root figure, and add the product to the third co-efficient; proceed thus to the end of the line, adding the last product to the absolute number. Again, multiply the first co-efficient by the root figure, and add the product to the sum under the second co-efficient; multiply the resulting sum by the root figure, and add the product to the sum under the third co-efficient; and so on, stopping under the last co-efficient. Repeat the process, stopping each succeeding time, one term nearer to the left hand, till the last sum falls under the second co-efficient. 5. Try how often the last sum under the last co-efficient is contained in the sum under the absolute number, and take the result for the next root figure. 6. Using the first co-efficient, and the last sum in each column, instead of the co-efficients and absolute number, proceed with this new root figure as with the preceding one. 7. Obtain another root figure in the manner last mentioned, and thus continue the operations as far as neces sary. Note 1.-In multiplying by each root figure, attention must be given to its value. Thus, if it is of the order of tens, the multiplication must be made by the number of tens which it represents; and so, for other values. Also, in the multiplications and additions, regard must be had to the signs of the numbers. 2. The signs of the successive sums, under the absolute number, must continue throughout the operation, the same as that of the absolute number. If the opera tion for any of the root figures cause the sign to change, a less value must be taken for that figure. This will not unfrequently occur with regard to the second root figure, but it will seldom be the case for the others. 3. After two or three root figures have been obtained, and the multiplications and additions corresponding to them have been completed, the succeeding parts of the operation may be contracted in the following manner. Cut off the right hand figure of the sum, in the column under the last co-efficient; the two right hand figures of the sum in the preceding column; the three right hand figures of the sum in the column preceding that, and so on. If either of the figures next to the right of the marks of separation is 5, or more than 5, add, mentally, a unit to the first figure on the left of the mark, when using it in the succeeding multiplication and addition. Repeat the same contraction for each of the following root figures. These contractions may commence, in cubic equations, after the second or third decimal figure in the root is obtained; in biquadratic equations, after the first or second decimal figure; and in higher equations, after the first decimal figure. And if the operation is closed when the sum under the absolute number is reduced to two figures, all the figures in the root will be true.* EXAMPLES. 1. Given 3x-4x2+2x-1000=0, to find the value of x. Ans. 4.342447603. This rule, improved from Young's Algebra, was communicated by my friend John Gummere, of Burlington. |