Page 2: Solving Polynomial Equations

This page expands on the process of finding pierwiastki całkowite i wymierne wielomianu (integer and rational roots of polynomials) through a detailed example.

The problem presented is to solve the equation x^3 - 6x^2 + 9x - 4 = 0.

Step 1: Find integer roots

• List divisors of -4: ±1, ±2, ±4
• Test each divisor in the polynomial

Highlight: After testing, we find that x = 1 and x = 4 are integer roots of the polynomial.

Step 2: Factor the polynomial

• Since x = 1 and x = 4 are roots, the polynomial is divisible by $x-1$ and $x-4$
• Perform polynomial long division to find the remaining factor

Example: $x^3 - 6x^2 + 9x - 4$ = x^2 - 5x + 4$$x - 1

This factorization helps in further analysis of the polynomial and its roots, demonstrating how to wyznacz pierwiastki wielomianu systematically.

Page 3: Completing the Solution

This final page focuses on solving the remaining quadratic equation to find all roots of the polynomial.

After factoring out $x-1$, we're left with the quadratic equation x^2 - 5x + 4 = 0.

Vocabulary: The quadratic formula is used to solve equations in the form ax^2 + bx + c = 0, where a ≠ 0.

Steps to solve the quadratic equation:

1. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 16 = 9
2. Apply the quadratic formula: x = $-b ± √Δ$ / (2a)

Example: x1 = $5 + √9$ / 2 = 4, x2 = $5 - √9$ / 2 = 1

The solution verifies our earlier findings that x = 1 and x = 4 are the only roots of the polynomial.

Highlight: This method demonstrates how to find pierwiastki wymierne wielomianu (rational roots of polynomials) by combining factoring and the quadratic formula.

This comprehensive approach showcases the Twierdzenie o pierwiastkach wymiernych (Theorem on Rational Roots) in action, providing a thorough understanding of how to wyznacz pierwiastki wielomianu in various scenarios.

Page 1: Theorem on Integer Roots of Polynomials

This page introduces the fundamental theorem for finding pierwiastki całkowite wielomianu (integer roots of polynomials).

The theorem states that if a polynomial w$x$ = anx^n + an-1x^n-1 + ... + a1x + a0 (a0 ≠ 0) with integer coefficients has integer roots, they will be divisors of the constant term a0.

Definition: Integer roots of a polynomial are whole number solutions that make the polynomial equal to zero when substituted for x.

An example is provided to demonstrate the application of this theorem:

1. Find the integer roots of w$x$ = 2x^3 - 3x^2 + 4x - 3

The process involves two main steps:

1. List the divisors of the constant term (-3 in this case): ±1, ±3
2. Test each divisor by evaluating the polynomial

Example: Evaluating w(1) = 2(1)^3 - 3(1)^2 + 4(1) - 3 = 2 - 3 + 4 - 3 = 0, confirming that 1 is a root.

This systematic approach efficiently narrows down potential integer roots, making it easier to wyznacz całkowite pierwiastki wielomianu.