CÓMO RESOLVER ECUACIONES DE TERCER GRADO. Método o regla de Ruffini

How to Solve Cubic Equations Using Ruffini's Method

Introduction to Cubic Equations

• The video introduces the topic of solving cubic equations using Ruffini's method, contrasting it with the well-known quadratic formula.

• It highlights that while there is a complex formula for cubic equations, Ruffini's method offers a simpler alternative.

Steps in Applying Ruffini's Method

• The first step involves organizing the coefficients of the polynomial from highest to lowest degree. For example, for x^3 - 2x^2 - 5x + 6, the coefficients are 1, -2, -5, and 6.

• The next step requires identifying potential rational roots by testing factors of the constant term (in this case, 6), including both positive and negative divisors: pm 1, pm 2, pm 3, pm 6.

Testing Potential Roots

• The presenter begins testing these candidates to find which one yields zero when substituted into the equation.

• After testing 1, it is confirmed as a root since substituting it results in zero. This indicates that x = 1 is a solution.

Continuing with Ruffini’s Method

• With one root found (x = 1), Ruffini’s method is reapplied to factor out this root from the original polynomial.

• A new synthetic division setup is created based on previous calculations to find additional roots.

Finding Additional Solutions

• The presenter tests another candidate root (-2) and confirms it as another solution after performing synthetic division.

• Finally, they test yet another candidate (3), confirming it as a third solution through similar steps.

Conclusion on Solutions Found

• By applying Ruffini’s method three times successfully, all solutions for the cubic equation are identified: x = 1, x = -2, x = 3.

Solving a Polynomial Equation Using Ruffini's Method

Introduction to the Problem

• The session begins with the introduction of a polynomial equation that needs solving, specifically using Ruffini's method.

• The polynomial is presented in descending order of degrees (3rd degree to 0th degree), and coefficients are identified: 1 (for x³), -5 (for x²), -17 (for x¹), and +21 (constant term).

Identifying Potential Solutions

• To find solutions, candidates must be numbers that divide the constant term (+21) exactly. Possible candidates include ±1, ±3, ±7, ±21.

• The speaker decides to test the candidate number 7 for potential roots of the polynomial.

Applying Ruffini's Method

• By applying Ruffini’s method with 7:

• The first coefficient (1) is brought down.

• Multiplying and adding steps yield results leading to a zero remainder.

• This confirms that x = 7 is indeed a solution since it produces a zero when substituted back into the polynomial.

Continuing with Further Solutions

• Next, Ruffini’s method is applied again using the remaining coefficients after factoring out (x - 7). Testing candidate number 1:

• Steps show that substituting x = 1 also yields a zero remainder.

• Thus, another solution found is x = 1.

Final Solution Discovery

• Finally, testing candidate number -3 through Ruffini’s method leads to another successful outcome:

• A zero remainder confirms that x = -3 is also a solution.

Conclusion of Findings

• The final solutions for the cubic equation are summarized as follows:

• Solutions found: x = 7, x = 1, and x = -3.