Overview
This lecture introduces polynomials, their structure and types, explains exponents, reviews the order of operations (PEMDAS), and demonstrates evaluating polynomial expressions.
Structure of Polynomials
- A polynomial is a sum or difference of terms.
- Each term consists of a coefficient (number), a variable, and possibly an exponent.
- All variable exponents in a polynomial must be whole numbers (0, 1, 2, ...).
- Variables cannot appear in denominators or have negative exponents.
Types of Polynomials
- A monomial has only one term.
- A binomial has two terms.
- A trinomial has three terms.
Understanding Exponents
- An exponent indicates how many times to multiply the base by itself (e.g., ( p^4 = p \times p \times p \times p )).
- Exponents provide a shorthand notation for repeated multiplication.
Order of Operations (PEMDAS)
- Use PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
- Parentheses include brackets and numerator/denominator expressions.
- Evaluate inside parentheses/brackets before other operations.
- Multiplication and division are performed as encountered from left to right.
- Addition and subtraction are performed as encountered from left to right.
Evaluating Polynomial Expressions (Examples)
- Substitute the given value for the variable before applying operations.
- In complex expressions, start with innermost parentheses, then exponents, then multiply/divide, finally add/subtract, working outwards.
- For fractions, treat the numerator and denominator as if they are in parentheses and evaluate separately before dividing.
Key Terms & Definitions
- Coefficient — The numerical factor in a term of a polynomial.
- Exponent — Indicates repeated multiplication of a base (variable).
- Polynomial — An expression consisting of the sum/difference of terms with whole number exponents.
- Monomial — A polynomial with one term.
- Binomial — A polynomial with two terms.
- Trinomial — A polynomial with three terms.
- PEMDAS — Mnemonic for the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Action Items / Next Steps
- Practice identifying and classifying polynomials as monomials, binomials, or trinomials.
- Complete assigned problems on evaluating polynomials using the order of operations.
- Review the rules for exponents and the PEMDAS order.