Polynomial and Rational Equations

This page introduces various równania wielomianowe przykłady (polynomial equation examples) and their solutions, demonstrating different techniques for solving polynomial and rational equations.

The page begins with a set of equations to solve:

Example: a) x³ + 8x² - 8x = -2 b) 2x⁵ - 64x² = 32x² c) -3x³ = 81 d) -2x³ = 40 e) -2x⁶ + 2 = 0

These examples showcase different degrees of polynomial equations, from cubic to sixth-degree polynomials.

Next, the page presents more complex equations:

Example: 1-2x$$3x+2x$4-5x$ = 0

This equation demonstrates the product of linear factors set to zero, a common form in równania wielomianowe zadania (polynomial equation problems).

The solution process involves finding the roots of each factor:

1. 1-2x = 0, x = 1/2
2. 3x+2 = 0, x = -2/3
3. x = 0
4. 4-5x = 0, x = 4/5

Highlight: The solution set is x ∈ {-2/3, 0, 1/2, 4/5}, showing that a polynomial equation can have multiple solutions.

The page concludes with additional examples of polynomial and rational equations, including:

Example: a) 3x + 2 = 0 b) 4 - 5x = 0 c) x² - 4 = 0 d) x³ - 1 = 0

These examples illustrate various metody rozwiązywania równań wielomianowych (methods for solving polynomial equations), including simple linear equations, quadratic equations, and cubic equations.

Vocabulary:

• Równanie wielomianowe: Polynomial equation
• Rozwiązanie: Solution
• Pierwiastek: Root

The page provides a comprehensive overview of równania wielomianowe jak rozwiązywać (how to solve polynomial equations), offering a range of examples that progress from simple to more complex equations, helping students build their problem-solving skills in this crucial area of algebra.