ECUACIONES POLINÓMICAS 1

Introduction to Polynomial Equations

Overview of Polynomial Equations

• The session focuses on solving polynomial equations, which are part of a broader category that includes trigonometric, logarithmic, linear, quadratic, and cubic equations.

• Before diving into polynomial equations, important definitions and theorems will be reviewed to establish foundational knowledge.

Definition of an Equation

• An equation is defined as an equality between two mathematical expressions separated by an equal sign. These expressions can include constants (e.g., numbers), variables (e.g., x, y, z), and operators (addition, subtraction, multiplication, division).

Examples of Different Types of Equations

• Examples provided include:

• Linear equation: x - 3 = 5

• Quadratic equation: x^2 + 3x - 4 = 0

• Cubic equation: x^3 - 6x^2 + 11x - 6 = 0

Solving Equations

Understanding Solutions

• To solve an equation means finding values for the variables that make the equality true. For instance:

• In the linear example x - 3 = 5, the solution is x = 8.

• For the quadratic equation x^2 + 3x - 4 = 0, solutions are x =1 or x =4.

• The cubic equation has three solutions: x =1, x =2, or x =3.

Roots and Solutions

Definitions of Roots

• Values that satisfy an equation are called solutions or roots. Specifically for single-variable equations like quadratics, these solutions are referred to as roots.

Algebraic vs. Polynomial Equations

Algebraic Equation Definition

• An algebraic equation encompasses a broader set of equations than polynomial ones; it includes operations such as addition, subtraction, multiplication, division with both positive and negative powers as well as roots.

Examples of Algebraic Equations

• Example one: 1/x -5 = r_a x

• Example two: x^-2 + x =0

Characteristics of Polynomials

Defining Polynomials

• A polynomial in variable x consists solely of positive integer exponents. When set equal to zero (e.g., a degree three polynomial equals zero), it forms a polynomial equation.

Key Features

• The degree must be greater than or equal to one.

• Coefficients can be any real number.

• A polynomial's general form is expressed as:

[

P(x)=a_n x^n + a_n−1 x^n−1 + ... + a_1 x + a_0

]

Conclusion on Polynomial Equations

Summary Insights

• A polynomial equation represents an equality involving polynomials set to zero.

• The degree n must be at least one for valid polynomials.

Understanding Polynomial Equations and Their Roots

Definition of Roots in Polynomials

• A number r is defined as a root or zero of a polynomial P(x) if substituting r into the polynomial results in zero: P(r) = 0 .

Solving Polynomial Equations

• To solve a polynomial equation P(x) = 0 , one must find values of x that make the polynomial equal to zero, effectively identifying its roots. This involves substituting found roots back into the polynomial.

Characteristics of Integer Roots

• If a root of a polynomial is an integer, it is specifically referred to as an integer root. The discussion includes examples illustrating this concept.

Types of Polynomial Equations

• Examples include:

• Linear equations like P(x) = Ax + b = c .

• Quadratic equations which are polynomials of degree two.

• All provided examples confirm they are indeed polynomials based on their structure.

Conditions for Being a Polynomial

• A valid polynomial must have non-negative integer exponents; negative or fractional exponents disqualify it from being classified as such. The degree must be greater than or equal to one with integer values (1, 2, 3, etc.).

Solving First-Degree Polynomials

• For first-degree polynomials, rearranging terms leads to solutions by isolating x . The formal method respects algebraic properties while providing clarity on how to manipulate equations correctly.

Example Solution Process for First-Degree Polynomials

• Steps include moving terms across the equality sign and simplifying until reaching the solution for x . This process illustrates both formal and informal methods for solving linear equations.

Verification of Solutions

• Substituting back into the original polynomial confirms whether calculated roots satisfy the equation, ensuring accuracy in finding roots through verification steps outlined in examples.

Exploring Second-Degree Polynomials

• An example demonstrates finding roots for second-degree polynomials using specific values (e.g., checking if r = 7 ). It emphasizes verifying that these values yield zero when substituted back into the equation.

Additional Example with Quadratic Solutions