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Algebraic equation of the nth degree of the general form

Lecture





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:

  Algebraic equation of the nth degree of the general form

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:

  Algebraic equation of the nth degree of the general form

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:

  Algebraic equation of the nth degree of the general form

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HANDBOOK ON MATHEMATICS, SCHOOL MATHEMATICS, HIGHER MATHEMATICS

Terms: HANDBOOK ON MATHEMATICS, SCHOOL MATHEMATICS, HIGHER MATHEMATICS