Polynomials Class Overview

Polynomials

• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Standard form: A polynomial is typically written in descending order of exponents.
• Examples: x² + 2x + 1, 3x³ - 2x + 5.

Types of Polynomials

• Monomial: A polynomial with only one term (e.g., 5x²).
• Binomial: A polynomial with two terms (e.g., x + 3).
• Trinomial: A polynomial with three terms (e.g., x² + 2x + 1).
• Degree: The highest power of the variable in a polynomial.
  • Example: The degree of 2x³ + 5x² - 3x + 1 is 3.

Operations with Polynomials

• Addition: Combine like terms (terms with the same variables and exponents).
• Subtraction: Combine like terms by essentially adding the additive inverse of the terms.
• Multiplication: Use the distributive property and combine like terms.
  • Example: (x+2)(x+3) = x(x+3) + 2(x+3) = x² + 3x + 2x + 6 = x² + 5x + 6.
• Division (using long division for the more complex cases): Use a process similar to division of whole numbers.

Factoring Polynomials

• Factoring is the reverse of multiplication. It involves rewriting a polynomial as a product of simpler polynomials.
• Techniques include:
  • Common factor: Identifying and factoring out the greatest common factor.
  • Grouping: Arranging terms in groups and factoring each group.
  • Difference of squares: Factoring expressions of the form a² − b².
  • Perfect square trinomials: Factoring trinomials that are perfect squares of binomials.
  • Sum/difference of cubes: Factoring expressions of the form a³ ± b³.

Remainder and Factor Theorems

• Remainder Theorem: If a polynomial f(x) is divided by (x - c), the remainder is f(c).
• Factor Theorem: A polynomial (x - c) is a factor of f(x) if and only if f(c) = 0.

Solving Polynomial Equations

• Set the equation equal to zero.
• Factor the polynomial.
• Use the zero product property to find the values that make each factor zero.
• Example: Solve x² + 2x - 3 = 0. Factoring gives (x+3)(x-1) = 0. The solutions are x = -3 and x = 1.

Roots of a Polynomial, Zeros, and x-Intercepts

• These terms all essentially refer to the values of x that make a polynomial equal to zero.
• Finding the roots / zeros / x-intercepts to a polynomial equation usually involves factoring, and using the remainder and factor theorems as needed.

Solving Quadratic Equations

• Standard form: ax² + bx + c = 0
• Methods:
  • Factoring: If factorable, this is the easiest method.
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This always works but can be more time-consuming on more complex quadratics.
  • Completing the square: Allows you to rewrite the equation in the form of a perfect square.

Graphing Polynomials

• The degree of a polynomial affects the general shape of the graph.
• X-intercepts are the roots/zeros.
• Y-intercept is the value of the polynomial when x = 0.
• Understanding the end behaviour (how the graph behaves as x tends to positive infinity and negative infinity) is also important.

Rational Polynomials

• A rational polynomial consists of two polynomials where one is in the numerator and the other is in the denominator.
• Rational expressions are commonly simplified by factoring the numerator and denominator.
• Operations (addition, subtraction, multiplication, division) are often required.
• Simplifying can involve factoring the numerator and denominator, then cancelling common factors.