Polynomial Division and the Remainder Theorem
This page covers essential concepts in polynomial division, including the Remainder Theorem and Bézout's Theorem.
The process of dzielenie wielomianów z resztą polynomialdivisionwithremainder is demonstrated through several examples. When dividing polynomials, the result consists of a quotient and a remainder, similar to integer division.
Example: For the polynomial Wx = x³ + x² - x + 5 divided by gx = x - 2, the division process is shown step-by-step, resulting in a quotient and remainder.
The Twierdzenie o reszcie z dzielenia wielomianu przez dwumian RemainderTheorem is introduced, which states that the remainder of a polynomial Wx divided by x−a is equal to Wa.
Highlight: The Remainder Theorem is a powerful tool for finding polynomial values and determining if a number is a root of a polynomial.
Several examples illustrate the application of the Remainder Theorem:
- Finding the reszta z dzielenia wielomianu remainderofpolynomialdivision Wx = -3x³ + Mx² + 8x - 6 by x+1, where M is determined using the theorem.
- Calculating the remainder when Wx = x³ + 4x² + 4x - 4 is divided by x+1.
- Determining the remainder of Wx = -2x³ + 10x² + x - 8 divided by x−2.
Vocabulary: Dzielenie wielomianów Horner Horner′smethod is an efficient algorithm for evaluating polynomials and performing polynomial division.
The page concludes with a mention of Twierdzenie Bezouta Beˊzout′sTheorem, which relates the roots of a polynomial to its factors. This theorem states that a number 'a' is a root of a polynomial if and only if x−a is a factor of the polynomial.
Definition: A root of a polynomial is a value that makes the polynomial equal to zero when substituted for the variable.
These concepts form the foundation for more advanced topics in polynomial algebra and are crucial for students to master for further mathematical studies.
