EQUAL ROOTS.] ALGEBRA 553 The divisors of 10, the last term, are 1, 2, 5, 10, each of which may be taken either positively or negatively; and these being substituted successively for x we obtain the following results: By putting +lfor.r, 1- 4-7 + 10= 0, -1, -1- 4 + 7 + 10= 12, + 2, 8- 16-14+10= -12, -2, -8- 10 + 14 + 10= 0, + 5, 125-100-35+10= 0. Here the divisors which produce results equal to are + 1, -2, and +5; therefore these are the three roots of the proposed equation SECT. XIV. SOLUTION OF EQUATIONS BY APPROXI MATION 114. When the roots of an equation cannot be accu rately expressed by rational numbers, it is necessary to have recourse to methods of approximation ; and by these we can always determine the numerical values of the roots to as great a degree of accuracy as we please. The application of methods of approximation is rendered easy by means of the following proposition : If two numbers, either whole or fractional, be found, which, when substituted for the unknown quantity in any equation, produce results with contrary signs, we may conclude that at least one root of the proposed equation is between those numbers, and is consequently real. Let the proposed equation be x 3 - 5x- + 10.r-15 = 0, which, by collecting the positive terms into one sum, and the negative into another, may also be expressed thus, then, to determine a root of the equation, we must find such a number as, when substituted for x, will render x*+Wx = 5x 2 + l5. .Let us suppose x to increase and to have every degree of magnitude from upwards in the scale of number ; then X A + lO.r and 5.e 2 + 15 will both continually increase, but with different degrees of quickness, as appears from the following table : Successive values of x ; 0, 1, 2, 3, 4, 5, 6, &c. -of x* + Wx; 0, 11, 28, 57, 104, 175, 276, &c. -- of 5u; 2 +15; 15, 20, 35, 60, 95, 140, 195, &c. By inspecting this table, it appears that while x increases from to a certain numerical value, which exceeds 3, the positive part of the equation, or x 3 + IQx, is always less than the negative part, or 5x~ + 15 ; so that the expression x 3 + Wx - (5x* + 15) or x 3 - 5.r 2 + lOu; - 15 must necessarily be negative. It also appears, that whrn x has increased beyond that numerical value, and which is evidently less than 4, the positive part of the equation, instead of being less than the negative part, is now greater, and therefore the expression a?-5i?+l Ox- 15 is changed from a negative to a positive quantity. Hence we may conclude that there is some real and determinate value of x, which is greater than 3, but less than 4, and which. Avill render the positive and negative parts of the equation equal to one another ; therefore that value of x must be a root of the proposed equation ; and as what has been just now shown in a particular case will readily apply to any equation whatever, the truth of the proposition is obvious. 115. From the preceding proposition it will not be difficult to discover, by means of a few trials, the nearest integers to the roots of any proposed numerical equation ; and those being found, we may approximate to the roots continually, as in the following example : x* -4o; 3 - 3*+ 27 = 0. To determine the limits of the roots, let 0, 1, 2, 3, 4, be substituted successively for x ; thus we obtain the fol lowing corresponding results : Substitutions for x, 0, 1, 2, 3, 4, Kesults, +27, + 21, + 5, -9, + 15. Hence it appears that the equation has two real roots, one between 2 and 3, and another between 3 and 4. That we may approximate to the first root, let us sup pose x = 2 + y, where y is a fraction less than unity, and therefore its second and higher powers small in com parison to its first power : hence, in finding an approxi mate value of y, they may be rejected. Thus we have J2y, &c. -4x 3 =-32-48y, &c. 3x = 6 3y + 27 =+27 Hence = 5 19 y nearly, and 3/= = 26; therefore, for a first approximation we have x =2 -2 6. Let us next suppose a; = 2 26+3/; then, rejecting as before the second and higher powers of y on account of their sniallness. and retaining three decimal places, we have y = -^- = -0075, and x = 2 26 + y = 2 2675. This value lo llU of x is true to the last figure, but a more accurate value maybe obtained by supposing x = 2 2675 +y", and pro ceeding as before. 116. The method we have hitherto employed for approxi mating to the roots of equations is known by the name of the method of successive substitutions, and was first proposed by Newton. It has been since improved by Lagrange, who has given it a form which has the advan tage of showing the progress made in the approximation by each operation. This improved form we now proceed to explain. Let a denote the whole number next less to the root sought, and - the fraction, which, when added to a, com pletes the root : then x = a + - . If this value of x be sub- y stituted in the proposed equation, a new equation involving y will be had, which, when cleared of fractions, will neces sarily have a root greater than unity. Let b be the whole number which is next less than that root ; then, for the first approximation, we have x = a + - . But b being only an approximate value of y, in the same manner as a is an approximate value of x, we may suppose y b + ; then, by substituting b + -, for y, we shall have a new equation, involving only y, which must be greater than unity. Putting therefore b to denote the next whole number less than the root of the equation involving y , we have y b + r,-) and substituting this value in that of x t the result is for a second approximate value of x. To find a third value, we may take y = b + , and so on, to obtain more accurate approximations.
70
