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5.1. POLYNOMIALS AND ALGEBRAIC EQUATIONS 161

3 . Ferrari solution.Let z 0 be any of the roots of the auxiliary cubic equation (5.1.4.3).
◦
Then the four roots of the incomplete equation (5.1.4.1) are found by solving the following
two quadratic equations:
q
2
y – √ z 0 y + p + z 0 + √ = 0,
2 2 z 0
q
2
y + √ z 0 y + p + z 0 – √ = 0.
2 2 z 0
TABLE 5.1
Relations between the roots of an incomplete equation of fourth-degree and the roots of its cubic resolvent

Cubic resolvent (5.1.4.3) Fourth-degree equation (5.1.4.1)
All roots are real and positive* Four real roots
All roots are real: one is positive and two are negative* Two pairs of complex conjugate roots
One real root and two complex conjugate roots Two real roots and two complex conjugate roots

5.1.5. Algebraic Equations of Arbitrary Degree and Their Properties
5.1.5-1. Simplest equations of degree n and their solutions.
1 .The binomial algebraic equation
◦
n
x – a = 0 (a ≠ 0)
has the solutions
⎧
⎪ 1/n 2kπ 2kπ
⎨ a cos + i sin for a > 0,
⎪
x k+1 = n n
⎩ |a| 1/n cos (2k + 1)π + i sin (2k + 1)π for a < 0,
⎪
⎪
n n
2
where k = 0, 1, ... , n – 1 and i =–1.
2 . Equations of the form
◦
n
x 2n + ax + b = 0,
n
x 3n + ax 2n + bx + c = 0,
n
x 4n + ax 3n + bx 2n + cx + d = 0
n
are reduced by the substitution y = x to a quadratic, cubic, and fourth-degree equation,
respectively, whose solution can be expressed by radicals.
Remark. In the above equations, n can be noninteger.
◦
3 .The reciprocal (algebraic) equation
2
a 0 x 2n + a 1 x 2n–1 + a 2 x 2n–2 + ··· + a 2 x + a 1 x + a 0 = 0 (a 0 ≠ 0)
can be reduced to an equation of degree n by the substitution
1
y = x + .
x

2
*Bythe Vi` ete theorem, the product of the roots z 1, z 2, z 3 is equal to q ≥ 0.