MAT137 - Week 8: Polynomials

Welcome to week 8 of MAT137! This week, we focused on polynomials, a cornerstone of algebra and calculus. We covered sections 6.1 to 6.5, building a solid foundation for future mathematical explorations.

Key Concepts Covered

  • Definition of a Polynomial: A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A general polynomial can be expressed as: $$P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants (coefficients) and $n$ is a non-negative integer (degree of the polynomial).
  • Polynomial Operations: We learned how to add, subtract, and multiply polynomials. Remember, when adding or subtracting, we combine like terms (terms with the same variable and exponent). Multiplication involves using the distributive property to multiply each term of one polynomial by each term of the other.
  • Polynomial Division: Understanding polynomial division is crucial. We covered both long division and synthetic division. Long division is a general method, while synthetic division is a shortcut applicable when dividing by a linear factor of the form $x - c$. When dividing a polynomial $P(x)$ by $(x-c)$, the result can be written as: $$P(x) = (x-c)Q(x) + R$$ where $Q(x)$ is the quotient and $R$ is the remainder.
  • The Remainder Theorem: A powerful tool! The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x-c)$, then the remainder is $P(c)$. This theorem provides a quick way to evaluate a polynomial at a specific value.
  • The Factor Theorem: The Factor Theorem is a direct consequence of the Remainder Theorem. It states that $(x-c)$ is a factor of $P(x)$ if and only if $P(c) = 0$. This helps us find roots of polynomials.
  • Finding Roots of Polynomials: We discussed strategies for finding roots (or zeros) of polynomials, including factoring, using the Rational Root Theorem, and applying the quadratic formula when applicable. The Rational Root Theorem provides a list of possible rational roots for a polynomial with integer coefficients.

Resources and Practice

To solidify your understanding of polynomials, be sure to:

  • Review the class notes from 11-03-25, covering chapters 6.1 to 6.5.
  • Complete the weekly quiz to test your knowledge.
  • Explore resources at Khanacademy.org for additional explanations and practice problems. They offer excellent tutorials on polynomials and related topics.

Polynomials are fundamental to many areas of mathematics, so a strong understanding here will benefit you greatly. Keep practicing, and don't hesitate to ask questions! Good luck!