Neglecting the terms in v3 and v2, we obtain as the approximation to the root of this equation .0002. That is, the approximation to the root of the given equation is 2.4142.

Since the figure in the fourth decimal place is less than 5, the required root, correct to three decimal places, is 2.414. The different steps need not be kept separate, but may be arranged compactly as follows:

It is not necessary to write out each new equation in order to find the first figure of the corresponding root. It is evident that this figure is, approximately, the first figure of the quotient obtained by dividing the last term of the equation by the

coefficient of the first power of the unknown number, neglecting sign.

If the coefficient of the first power of the unknown number be 0 at any stage, the corresponding figure of the root is obtained by dividing by the coefficient of the second power of the unknown number, and taking its square root. For, the approxi

mate equation is then of the form my n = 0, or y =

√

n

m

60. The example of the preceding article illustrates the following method:

Find the integral part of the root by the method of Art. 53. Form the equation whose roots are less than the roots of the given equation by this number.

Find the next figure of the root by dividing the last term of the transformed equation by the coefficient of the first power of the unknown number, neglecting sign, and checking the first figure of the quotient.

Form the equation whose roots are less by this number than the roots of the first transformed equation.

Take as the next figure of the root the first figure of the quotient obtained by dividing the last term of the second transformed equation by the coefficient of the first power of the unknown number; and so on.

The last term of each transformed equation must have the same sign as the last term of the given equation.

If, for any transformed equation, too great a figure be obtained by the required division, the sign of the last term of the next transformed equation will be opposite to that of the given equation.

If, for any transformed equation, too small a figure be obtained by the required division, the number suggested for the next decimal place will be greater than 9.

61. In applying Horner's Method, the decimal point can be avoided as follows, referring to the example of Art. 59.

The decimal point is about to appear in the second stage of the work, the root of the first transformed equation being .abcd.

Before finding the value of a, let us form the equation whose roots are 10 times the roots of this equation, as in Art. 21. The coefficients of the resulting equation are 1, 70, 900, -5000.

The corresponding root of this equation is a.bcd. We now have 5000 = 5. As in Art. 59, we find that the root is less than 5 and greater than 4; that is, is 4.bcd.

Diminishing the roots by 4, we obtain a transformed equation whose corresponding root is .bed.

216000

As before, we multiply the roots of this equation by 10, and obtain as the next figure of the root, 1, 116888, approximately. The work follows:

=

In general, when the decimal point is about to appear, annex one cipher to the second coefficient (counting from the left), two to the third, and so on. Proceed as in Art. 59, noting that the synthetic divisor is then an integer.

Proceed in like manner with each transformed equation.

62. A negative irrational root is obtained by first transforming the equation to one whose roots are those of the given equation with signs changed, and finding the corresponding positive root of this equation.

Roots of Numbers.

63. An approximate value of any real root of any number can be obtained by Horner's Method.

The q roots of this equation are the q qth roots of n.

EXERCISES X.

1-14. Find the irrational roots, correct to three decimal places, of each of the equations in Exercises IX., Exx. 1-14.

The following equations have each two imaginary roots. Find the real rational roots; remove them and find the irrational roots of the depressed equation. Remove these roots, and solve the final depressed equation for the imaginary roots:

15. x 11 x3 + 38 x2 - 51 x + 27 = 0.

Each of the following equations has two rational roots which can be expressed as decimal fractions. Find them, by Horner's Method, and solve the depressed equation:

Find, correct to three decimal places, by Horner's Method, the values of the following arithmetical roots :

22. /9. 23. 3/2.5. 24. 1/8.

25. 9.2.

26.5.7.

Sturm's Theorem.

64. Descartes' Rule does not give the exact number of positive and negative roots of an equation. The principle of Art. 55 does not inform us definitely whether there is one or an odd number of roots between a and b, when ƒ(a) and ƒ(b) have opposite signs; whether there is no root or an even number of roots between a and b, when ƒ(a) and ƒ(b) have the same sign. The following theorem, discovered by Sturm, a Swiss mathematician, in 1829, gives the exact number of real positive and negative roots of an equation, and also the exact number which lie in any interval. Before applying this theorem, it is assumed that all the multiple roots, if any, have been obtained and the depressed equation formed. Let f(x)=0 be this depressed equation.

Then, f(x) and f'(x) have no common factor. In the method now to be employed, the process of finding the H. C. F. of two expressions is continued until a remainder free of x is obtained.

But the sign of each remainder is changed before it is used in the next stage as a divisor.

Let fi(x), f2(x), etc., be the remainders, with signs changed.

Then the functions, f(x), f'(x), fi(X), ƒ2(X), .....

are called Sturm's Functions. We now have

ƒ(x) = Q1f'(x) +R1, or since R1 = − ƒ1(x),

65. The following principles are derived from the relations of Art. 64 : (i.) Two consecutive functions cannot reduce to 0 for the same value of x.

Suppose f(x) and fs(x) reduce to 0 for the same value of x; that is, have a common factor of the form x r. Then, from equation (3), f(x) and f(x) reduce to 0 for this value of x, and have a common factor xr. In like manner, from (2), we infer that f'(x) and si(x), and finally, from (1), that ƒ(x) and ƒ'(x) reduce to 0 for this value of x; that is, have a common factor x-r. But this is contrary to the hypothesis that f(x) and ƒ'(x) do not have a common factor.

(ii.) Any value of x which reduces to 0 any function except the first gives to the two adjacent functions opposite signs.