Polynomials

Question 1

Defenition of a polynomial.

Answer 1

Polynomial p is a real-valued function p(x)=a₀+a₁x²+…+a_nxⁿ=Σa_ixⁱ defined on R, where “a” is areal constant(coefficient) and “n” is a degree of p(x).

Question 2

Defenition of roots of polynomials.

Answer 2

a_nxⁿ+a_n-1x^n-1+…+a₁x+a₀=0 is algebraic equation of n-th degree, where “a” are real coefficients, a_n/=0 . By root of this AE we mean any real or complex number ð wich satisfies the equation when substitute x=ð

Question 3

Theorem about polynomial’s roots(ð=r/s)

Answer 3

Let ð=r/s be a rational number written in the irreducible form. Let p(x)=Σa_jx^{<span>j </span>}where “a” is integrs and a₀/=0 and a_n/=0. If p(ð)=0, then r and s must satisfy following condicoins:

• a₀ is divisible by r and a_n is divisible by s
• the number p(1) (resp, p(-1)) is divisible by r-s(resp r+s)

Question 4

Theorem about complex conjugate (Komplexně sdružené číslo)

Answer 4

Let p(x) be a real polynomial and let ð be a complex root of p(x). Then, the complex conjugate of ð is also a root of p(x).

• Consequence: each polynomial of odd(3) degree has atleast 1 real root.

Question 5

Theorem about polynomials roots and its degree.

Answer 5

Let p(x) be a polynomial and let degP(x)=n, then p(x) has at most n mutualy distinct roots.

Question 6

Defenition of multiple roots.

Answer 6

Let p(x) be a polynomial and let ð be a root of p(x). Let us say, that ð is a root of multiplicity k, k(k€N), if p(x) is divisible by the polynomial (x-ð)^k, but not divisible by polynomial (x-ð)^k+1 .

Question 7

Defenition of irreducible polynomial.

Answer 7

Polynom p(x) je irreducibilni pokud jej neni mozne rozlozit na soucin polynomu r(x), s(x) stupne aspon 1. Pro irreducibilni polynom neplati p(x)=r(x)*s(x)