Introduction

2.1 Introduction

Description

Quick Overview

This section provides a foundational understanding of polynomials, focusing on their degree and classifying them into linear, quadratic, and cubic categories.

Standard

In this section, students revisit their prior knowledge of polynomials, particularly their degrees and types, including linear, quadratic, and cubic polynomials. Key examples and definitions are introduced to clarify the classification of polynomials based on their degree, with an emphasis on the importance of zeroes in their equations.

Detailed

Detailed Summary

In this section, we reiterate the concept of polynomials, emphasizing that the highest power of a variable in a polynomial determines its degree. A polynomial of degree 1 is known as a linear polynomial (e.g., 2x - 3), while a degree 2 polynomial is termed a quadratic polynomial (e.g., x² - 3x - 4). Likewise, a cubic polynomial is of degree 3 (e.g., x³ - x²). The section explains the significance of zeroes in polynomials, noting that substituting a value for x to yield zero identifies the polynomial's zeroes. Graphical representations are introduced to visualize the zeroes of both linear and quadratic polynomials, laying the groundwork for understanding their geometrical meanings and how they relate to their coefficients.

Key Concepts

  • Polynomial: An algebraic expression of variables and coefficients.

  • Degree: The highest power of the variable in the polynomial.

  • Types of Polynomials: Linear, Quadratic, Cubic.

  • Zeroes: The x-values where a polynomial equals zero.

Memory Aids

🎵 Rhymes Time

  • To find zeroes that shine, set your polynomial to line.

📖 Fascinating Stories

  • Once upon a time, a polynomial wanted to find its roots. It met others like Linear, Quadratic, and Cubic, each with a story about their degrees.

🧠 Other Memory Gems

  • LQ for Linear and Quadratic, C for Cubic in our cubic catalog - remember the degrees well!

🎯 Super Acronyms

Remember Z stands for Zeroes where p(x) equals zero!

Examples

  • Example of a linear polynomial: p(x) = 2x + 3, degree = 1.

  • Example of a quadratic polynomial: p(x) = x² - 3x + 2, zeroes found by factoring.

  • Example of a cubic polynomial: p(x) = 3x³ - 3x² + x, degree = 3.

Glossary of Terms

  • Term: Polynomial

    Definition:

    An algebraic expression that consists of variables and coefficients, combined using only addition, subtraction, multiplication, and whole number exponentiation.

  • Term: Degree of a Polynomial

    Definition:

    The highest power of the variable in a polynomial.

  • Term: Linear Polynomial

    Definition:

    A polynomial of degree 1.

  • Term: Quadratic Polynomial

    Definition:

    A polynomial of degree 2.

  • Term: Cubic Polynomial

    Definition:

    A polynomial of degree 3.

  • Term: Zero of a Polynomial

    Definition:

    A value of x for which the polynomial evaluates to zero.