Algebraic equation of the nth degree of the general form

Types of algebraic equations



a _n x ⁿ + a _n-1 x ^n-1 + ... + a ₁ x + a ₀ = 0. - Algebraic equation of the n-th degree of general form.

Let the coefficients a _k be real or complex numbers.

1. For brevity, we denote the left side of the equation, which is a polynomial of degree n, as follows:



The number x = ξ is called the root of the equation, as well as the root of the polynomial P _n (x), if P _n ξ = 0. The number x = ξ is called the root of multiplicity m, if P _n (x) = (x = ξ) ^m Q _nm (x), where m is a positive integer, 1 ≤ m ≤ m and Q _nm (x) is a polynomial of degree n - m, such that Q _nm (ξ) 0.

2. The main theorem of algebra . The algebraic equation of nth degree has exactly n roots (real or cosplex), and the roots of multiplicity m occur exactly m times.

3. If an algebraic equation has roots x ₁ , x ₂ , ..., x _s , of multiplicities k ₁ , k ₂ , ..., k _s (k ₁ + k ₂ + ... + k _s = n), then the left side of the equation can be represented as:



4. An algebraic equation of odd degree with real coefficients always has at least one real root.

5. Suppose an algebraic equation with real coefficients has a complex root ξ = α + iβ. Then this equation must also have a root η = α - iβ, and the multiplicities of both roots are the same. 6. The algebraic equation of degree n with integer coefficients a _k cannot have other rational roots than non-contractible fractions p / q, moreover p is the divisor of a ₀ and q is the divisor of a _n . If a _n = 1, then all the rational roots of an algebraic equation are the integral divisors of the free a ₀ and can be easily found.

7. Any equation of degree ≤ 4 is resolvable in radicals, which means that its roots can be expressed using the operations of addition, division, subtraction and multiplication, as well as extraction of the root. For n ≥ 4, algebraic equations, basically, in radicals are insoluble. This statement is called the Ruffini-Abel theorem.

8. The Viet theorem . The following relations between the roots of an algebraic equation (taking into account their multiplicity) and their coefficients take place: