shall merely notice the existence of large classes of equations of the superior orders, which admit of such resolution into factors, and consequently of complete solution.

Since x3-1 = (x − 1) (x2 + x + 1), for all values of x (Art. 69, Ex. 5), it follows that

x3- 1 = 0,

when x - 1 = 0, or when x2 + x + 1 = 0.

1, which is the arith

The first equation, x 10, gives a = metical root (Art. 645,) of the primitive equation. The second equation, or x2+ x + 1 = 0, gives

x =

− 1 ± √3 √ − 1,

2

The cube

roots of 1.

or two unarithmetical and imaginary roots, which also satisfy the primitive equation*.

Inasmuch as the equation 3-10 gives a = 1, and thereforex = 3/1, it follows that there are three cube roots of 1, which

The square of one of these imaginary roots is equal to the other: for

1-2√3√-1-3 1-√3

2

=

=

If, therefore, a represents one of these cube roots, a2 will represent the other, and the three cube roots of 1 may be represented by 1, a, a: or they may be represented by a, a2, a3, since a3 = 1.

where a

quently the three cube roots of 1 may be represented by 1, a and, is one of the imaginary cube roots of 1.

Every integral power of a cube root of 1 is a cube root of 1.

For every integral power of 1 is 1: and since all numbers are expressible by the formulæ 3n, 3n + 1, 3n + 2 (Art. 526, Note), where n is a whole number, it follows that

The cube

roots of -1.

The properties of the cube and higher roots of 1 are connected with many important theories, and will be more particularly considered in a subsequent Chapter, (xxIII).

670. To solve the equation

23+1 = 0.

Since x3 + 1 = (x + 1) (x2 − x + 1), for all values of x, it follows that

The sixth roots of 1.

2

Inasmuch as the equation, a3 + 1 = 0, gives x3 = forex = 1, it follows that the three cube roots

- 1, and thereof — 1, are

For x − 1 = (x3 − 1) (x3 + 1) for all values of x, and the roots of x-1=0 (Art. 668,) are therefore those of

The same expression a-1 may be likewise resolved into the three quadratic factors

* If a be one of these imaginary roots, the three roots may be expressed

by · 1, a and a2, or by − 1, a and, or by a, a2 and a3: for a3 = − 1.

672. Biquadratic equations, which present themselves under Solution of

and consequently admit of solution by the processes given for quadratic equations: for if we make au, the equation (1) becomes

which become the factors above mentioned, when u is replaced by x2.

The four roots of the proposed biquadratic equation are, therefore,

biquadratic equations wanting the second and fourth

terms of the general form in

Art. 667.

roots exhi

673. The roots of the equation considered in the last Article, The same may be exhibited under other and equivalent forms, which are bited under sometimes susceptible of more complete arithmetical reduction. a different For if we divide the original equation

form.

If we multiply both sides of this equation by x, we get

and therefore

x2 + √s=±√(2 s−q) x,

x2 = √(2 √s − q) x = − √s.

Solving this pair of quadratic equations in the ordinary manner, we get

x=

·√(√3 - 2) + √(√5 - 2) •,

where the different combinations of the signs + and -1 will furnish the four different roots of the primitive equation (1).

* If we compare the equivalent expressions for a which are obtained in Arts. 672 and 673, we find

√ { - { ± √ (ï − s) } = √(√3 − 1) ± √ ( − ¥3 − 2),

and, consequently,

may

--

be considered as expressing the square roots of -±√(-s),

result which may be easily verified: and if we replace

therefore s by a- b, we shall find

2 by

a,

a

-s by b, and

4

which assumes the form a± √ẞ in those cases in which a-b is a com

resolvible

675. If we extract the square root of the first member Cases of biquadratic of a biquadratic equation arranged as in Art. 667, (where all equations its significant terms are transposed to one side,) and if we arrive which are at a remainder, which, with its sign changed, is either a complete into factors by the prosquare or independent of the unknown symbol, the biquadratics of equation may, in all cases, be resolved into two quadratic factors, extracting and its roots determined by the solution of two quadratic equa- root of tions.

the square

their first member as

Art. 667.

For if we express by X the first member of the equation, arranged in and by u the terms which are obtained in the root before the process of extracting its square root terminates, then X-u2 is the remainder: and if

we have

and this is

=

X-u2=- v2,

X = u3 − v2 = (u + v) (u − v),

= 0, when u + v = 0, or when u v = 0 (Art. 668): the roots of the equations u + v = 0 and u - v = 0 are therefore the roots of the equation X = 0.