Unit 1: Advanced Algebra

Section B: Polynomials and Factoring Techniques

Welcome

Welcome to Section B: Polynomials and Factoring Techniques! In this section, you’ll explore polynomials, which are algebraic expressions that consist of variables and coefficients, and learn how to manipulate and factor them for problem-solving.

Imagine

Imagine you’re working on designing a roller coaster, where the track’s shape can be modeled by polynomial equations. Understanding how to factor and manipulate these equations is crucial in ensuring the ride is safe and thrilling.

Context

Previously, you’ve studied quadratic functions. Now, we’ll extend those ideas to polynomials of higher degrees and explore various methods for factoring them to solve complex equations.

Overview

This section covers polynomial functions, adding, subtracting, and multiplying polynomials, special products, and advanced factoring techniques. You’ll learn how to apply these concepts to solve equations and model real-world scenarios.

Objectives

  • Understand the properties and characteristics of polynomial functions.
  • Perform operations on polynomials, including addition, subtraction, and multiplication.
  • Identify and apply special product formulas, such as the difference of squares and perfect square trinomials.
  • Use various factoring techniques to solve polynomial equations.
  • Apply polynomial functions to real-world problems, such as in engineering and physics.

Preparatory Guidance

Definitions and Pronunciations
  • Polynomial: An algebraic expression that consists of terms in the form ax^n, where a is a coefficient and n is a non-negative integer.
  • Degree: The highest power of the variable in a polynomial.
  • Factorization: The process of breaking down a polynomial into simpler factors that can be multiplied to obtain the original polynomial.
Verbal Reading of Equations
  • For P(x) = ax^n + bx^{n-1} + \ldots + c, say “P of x equals a x to the n plus b x to the n minus one plus c.”
  • For x^2 - y^2 = (x+y)(x-y), say “x squared minus y squared equals x plus y times x minus y.”
Problem-Solving Strategies
  • Identify and analyze polynomial functions, using operations to combine and simplify them.
  • Apply special product formulas, such as the difference of squares, to factor and simplify polynomial expressions.
  • Use factoring techniques, such as grouping and synthetic division, to solve polynomial equations.
  • Explore real-world applications of polynomial functions in engineering, physics, and other fields.
  • Use graphing calculators or software to visualize polynomial functions and their factors.

Considerations

How do polynomials model real-world phenomena? Why is it important to understand factoring techniques for solving polynomial equations? In what ways can you apply polynomial functions in your daily life or future career?