2.2 Polynomials in One Variable

Description

Quick Overview

The section provides an overview of polynomials in one variable, introducing key definitions, types, and properties, including degrees and coefficients.

Standard

This section discusses polynomials in one variable, detailing their components such as terms, coefficients, and degrees. It classifies polynomials based on the number of terms and their degrees, further explaining the significance of zeroes in polynomial equations. The section concludes by emphasizing the importance of polynomials in algebraic expressions and their applications.

Detailed

Detailed Summary

In this section, we explore polynomials in one variable, defined as algebraic expressions of the form:

Polynomials

Where:
- a₀, a₁, a₂, ..., aₙ are constants (coefficients)
- x is the variable
- n is a non-negative integer (degree of the polynomial)

The expression is considered a polynomial if every term is of the form c * (variable) with whole number exponents. We categorize polynomials based on the number of terms:
- Monomial: A polynomial with one term (e.g., 5x).
- Binomial: A polynomial with two terms (e.g., x + 2).
- Trinomial: A polynomial with three terms (e.g., x² + 3x + 2).

The degree of a polynomial is the highest power of the variable present. For example, in the polynomial 4x^3 + 7x^2 - x + 1, the degree is 3. The section further highlights zeroes of polynomials, explaining that a zero of p(x) is a real number c such that p(c) = 0. The importance of the Factor Theorem and its application in identifying factors of polynomials based on their zeroes is also discussed, along with the use of polynomial equations in algebraic problem solving.