Therefore, the transformed equation is

5y+28y3+51y2+32y-1=0.

This laborious operation can be avoided by the synthetical method of division (Art. 312).

Taking the same example, and recollecting that in the synthetical method, the first term of the divisor not being used, may be omitted, and that the first term of the quotient, by which the modified divisor is to be multiplied for the first term of the product, is always the first term of the dividend; the whole of the work may be thus arranged:

for it is plain that the first remainder will fall under the absolute term, the second under the term next to the left, and so on. Hence, the transformed equation is

5y+28y3 + 51y2 + 32y − 1 = 0.

2. Find the equation whose roots are less by 1.7 than those of the equation

First, find an equation whose roots are less by 1.

We have thus found the co-efficients of the terms of an equation whose roots are less by 1 than those of the given equation: the equation is

x3 + x2+2x-2=0;

and now by finding a new equation whose roots are less than those of the last by .7, we shall have the required equation: thus,

hence, the required equation is

y3+3.1y24.87y+.233 0.

=

This latter operation can be continued from the former, without arranging the co-efficients anew. The operations have been explained separately, merely to indicate the several steps in the

transformation, and to point out the equations, at each step resulting from the successive diminution of the roots. Combining the two operations, we have the following arrangement:

We see, by comparison, that the above results are the same as those obtained by the preceding operations.

3. Find the equation whose roots shall be less by 1 than the roots of

237x+7=0.

Ans. y3+3y2—4y+ 1 = 0.

4. Find the equation whose roots shall be less by 3 than the roots of the equation

Ans. y9y3 + 12y2 — 14y = 0.

5. Find the equation whose roots shall be less by 10 than the roots of the equation

x4 + 2x3 + 3x2 + 4x 12340 0.

Ans. y+42y3+663y+4664y = 0.

6. Find the equation whose roots shall be less by 2 than the roots of the equation

Ans. y510y++ 42y+86y2 + 70y + 4 = 0.

Horner's Method of approximating to the Real Roots of Numerical Equations.

314. The method of approximating to the roots of a numeri cal equation of any degree, discovered by the English nathe matician W. G. Horner, Esq., of Bath, is a process of very remarkable simplicity and elegance.

The process consists, simply, in a succession of transforma tions of one equation to another, each transformed equation, as it arises, having its roots equal to the difference between the true value of the roots of the given equation, and the part of the root expressed by the figures already found. Such figures of the root are called the initial figures. Let

be any equation, and let us suppose that we have fou ́ a part of one of the roots, which we will denote by m, and denote the remaining part of the root by r.

Let us now transform the given equation into another, whose roots shall be less by m, and we have (Art. 313),

V' = pm + P'pm- 1 + Q'pm 2 . . . . + T'r + U′ = 0 - - (2). Now, when r is a very small fraction, all the terms of the second member, except the last two, may be neglected, and the first figure, in the value of r, may be found from the equation

The first figure of r is the first figure of the quotient obtained by dividing the absolute term of the transformed equation by the penultimate co-efficient.

lf, now, we transform equation (2) into another, whose roots shall be less than those of the previous equation by the first figure of r, and designate the remaining part by s, we shall have,

V"s+ Psm-1+Q'sm-2.... + T''s +"= 0,

the roots of which will be less than those of the given equa tion by m the first figure of r. The first figure in the value of s is found from the equation,

U"

T

We may thus continue the transformations at pleasure, and each one will evolve a new figure of the root. Hence, to find the roots of numerical equations.

I. Find the number and places of the real roots by Sturms' theorem, and set the negative roots aside.

II. Transform the given equation into another whose roots shall be less than those of the given equation, by the initial figure or figures already found: then, by Sturms' theorem, find the places of the roots of this new equation, and the first figure of cach will be the first decimal place in each of the required roots.

III. Transform the equation again so that the roots shall be less than those of the given equation, and divide the absolute term of the transformed equation by the penultimate co-efficient, which is called the trial divisor, and the first figure of the quotient will be the next figure of the root.

IV. Transform the last equation into another whose roots shall be less than those of the previous equation by the figure last found, and proceed in a similar manner until the root be found to the required degree of accuracy.

REMARK I. This method is one of approximation, and it may happen that the rejection of the terms preceding the penultimate term will affect the quotient figure of the root. To avoid this source of error, find the first decimal places of the root, also, by the theorem of Sturm, as in example 4, page 399, and when the results coincide for two consecutive places of decimals, those subsequently obtained by the divisors may be relied on.

REMARK II.--When he decimal portion of a negative root is to be found, first transform the given equation into a.other by changing the signs of the alternate terms (Art. 280), and then find the decimal part of the corresponding positive root of this new equation.