a2 + pa3 + qa2+pa+1=0;

1

but, by substituting in place of x, it becomes

or

1

1

+p+q+p+1=0;

1+pa+qa2+pa3+a1=0;

which is nothing more than the preceding equality, written in an inverse order.

Hence we see that the roots of these equations, taken two and two, are the inverse or reciprocals of each other; whence it follows, that when half of the roots are determined, the other half can be obtained by dividing unity by each of the first. We will now show that the resolution of every reciprocal equation can be reduced to that of an equation of a subduple degree. Take the equation of the 6th degree, contashay

x® +px5+qx1 +rx3 +qx2 +px+1=0;

dividing it by x3 (3 being half of the degree of the equation), it takes the form

the same value of z; we can therefore obtain the values of x when those of z are known.

1

From the equation +=z, we deduce successively

x

2d. by multiplying the two new equations together,

z3—3z+p(z2 —2)+qz+r=0;

or reducing, z3 +pz2 +(q−3)z+r−2p=0;

an equation of the 3d degree, whilst the proposed equation is of the 6th. In like manner we might reduce an equation of the 4th, 8th, 10th, 12th....degree, to one of the 2d, 4th, 5th, 6th.... degree.

It yet remains to consider reciprocal equations of an uneven degree.

Take, for example, the equation

x3 +px1 +qx3 +qx2 +px+1=0.

We see at once that -1 is a root of this equation, for if we substitute-1 for x, we will obtain for the result —1+p−q+q−p+1, an expression in which all the terms destroy each other. (It will be the same for every reciprocal equation of an uneven degree.)

Hence (No. 252) the first member is divisible by x+1; and by performing this division we obtain (No. 253),

a reciprocal equation of an even degree, upon which we can operate as in the case treated of above.

301. N. B. The roots of an equation of an uneven degree are also the reciprocals of each other, taken two and two, when the coefficients of the terms at equal distances from the extremes, are equal and affected with contrary signs.

Take, for example, the equation

or, clearing the fraction and changing the signs,

x2+px1+qx3—qx2 —px—1=0.

1

Hence, a being one of the roots of this equation, is neces

sarily another.

Moreover, it is visible that 1 will verify the equation; and if we divide by x-1, it becomes

an equation in which the coefficients at equal distances from the extremes are equal and affected with the same sign.

302. Applications. Let there be the general equation involving two terms; it is evident that 1 is a root of it; and dividing by x-1, we find (No. 31)

x-1+x+x2-3+...+x2+x+1=0,

a reciprocal equation, the resolution of which can, by means of the preceding principles, be reduced to that of an equation of a subduple degree.

Take, for example, the equation

by dividing by x-1, we have

x2+x3+x2+x+1=0,

which can be put under the form

1

Making x+=z, whence x-zx+1=0; there will result

in the equation

Substituting the values of x+, and x2+

hence, by substituting for z its two values, and reducing, we

The equation 1o-1=0, can also be completely resolved by the same method.

For, we have

!o−1=(2*−1)(z+1)=0.

We already know the roots of the equation x-1=0; as to those of the equation x5 +1=0, since, by changing x -x, it becomes 5-10, it is only necessary to take the roots of this equation with contrary signs.

into

a

CHAPTER VII.

Resolution of Numerical Equations, involving one or more Unknown Quantities.

THE principles established in the preceding chapter, are applicable to all equations, whether their coefficients are numerical or algebraic, and these principles should be regarded as the elements which have been employed in the resolution of equations of the higher degrees.

It has been said already, that analysts have hitherto been able to resolve only the general equations of the third and fourth degree. The formulas they have obtained for the values of the unknown quantities are so complicated and inconvenient, when they can be applied, (which is not always possible,) that the problem of the resolution of algebraic equations, of any degree whatever, may be regarded as more curious than useful. Therefore analysts have principally directed their researches to the resolution of numerical equations, that is, to those which arise from the algebraic translation of a problem in which the given quantities are particular numbers; and methods have been found, by means of which we can always determine the roots of a numerical equation of any given degree.

It is proposed to develope these methods in the first part of this chapter.

The object of the second part is the supplement of elimination, or the resolution of numerical equations involving two or more unknown quantities.

To render the reasoning general, we will represent the pro