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 (polynomial division with remainder) 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 W(x) = x³ + x² - x + 5 divided by g(x) = 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 (Remainder Theorem) is introduced, which states that the remainder of a polynomial W(x) divided by (x - a) is equal to W(a).
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 (remainder of polynomial division) W(x) = -3x³ + Mx² + 8x - 6 by (x + 1), where M is determined using the theorem.
- Calculating the remainder when W(x) = x³ + 4x² + 4x - 4 is divided by (x + 1).
- Determining the remainder of W(x) = -2x³ + 10x² + x - 8 divided by (x - 2).
Vocabulary: Dzielenie wielomianów Horner (Horner's method) is an efficient algorithm for evaluating polynomials and performing polynomial division.
The page concludes with a mention of Twierdzenie Bezouta (Bézout's Theorem), 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.
