Schemat Hornera: A Comprehensive Overview

The Schemat Hornera is a fundamental technique in algebra, particularly useful for dzielenie wielomianów (polynomial division). This page provides a detailed explanation of the method, along with several examples to illustrate its application.

Definition: The Schemat Hornera is an algorithm used for dividing polynomials by linear factors of the form (x - c), where c is a constant.

The method involves arranging the coefficients of the polynomial W(x) in a specific manner and performing a series of calculations to obtain the coefficients of the quotient polynomial and the remainder.

Example: For the polynomial (x² + 5x - 1) divided by (x - 1), the Schemat Hornera can be applied as follows: 1 | 5 | -1 1 | 6 | 5 This results in the quotient (x + 6) with a remainder of 5.

The page also presents more complex examples, such as dividing (-3x³ + x² - 5x + 1) by (x - 1), demonstrating the versatility of the Schemat Hornera method.

Highlight: The Schemat Hornera can be used to find both the quotient polynomial and the remainder in a single process, making it more efficient than traditional polynomial long division.

Several practice problems are provided, allowing readers to apply the Schemat Hornera technique to various polynomial divisions. These examples range from simple second-degree polynomials to more complex higher-degree polynomials.

Vocabulary:

• Współczynniki: Coefficients
• Reszta z dzielenia: Remainder from division
• Wielomian: Polynomial

The page concludes with a comprehensive example demonstrating how to use the Schemat Hornera to divide (4x³ - x² - 10) by (2x - 6), illustrating the method's applicability to polynomials with non-unit leading coefficients.

Quote: "W(x) = G(x)F(x) + R(x)" - This fundamental equation summarizes the relationship between the dividend W(x), divisor G(x), quotient F(x), and remainder R(x) in polynomial division.

Understanding and practicing the Schemat Hornera is essential for students studying advanced algebra and those preparing for mathematical competitions or higher-level mathematics courses.