Overview
This lecture introduces polynomials, explains their terms, classifications (monomial, binomial, trinomial), degree, standard form, and key vocabulary relevant to polynomials.
What is a Polynomial?
- A polynomial is an expression made up of the sum of terms, each with a coefficient and a variable to a nonnegative integer power.
- Example: 10x⁷ − 9x² + 15x³ + 9 is a polynomial.
- Polynomials can have any number of terms, including just a single constant or variable term.
Terms and Coefficients
- Each part separated by + or − in a polynomial is called a term.
- Each term consists of a coefficient (the number multiplying the variable) and the variable raised to a nonnegative integer power.
- The coefficient can be any real number (positive, negative, integer, fraction, or irrational).
Types of Polynomials: Monomial, Binomial, Trinomial
- A monomial has one term (e.g., 6, 10z¹⁵, πb⁵).
- A binomial has two terms (e.g., 3y³ + 5y).
- A trinomial has three terms (e.g., 7y² − 3y + π).
Degree of Polynomials
- The degree of a term is the exponent of the variable in that term.
- The degree of a whole polynomial is the highest degree among its terms.
- Constants are considered zero-degree terms (e.g., 7 = 7x⁰).
Standard Form, Leading Term, and Leading Coefficient
- A polynomial in standard form arranges terms in order of decreasing degree.
- The leading term is the first term in standard form (highest degree).
- The leading coefficient is the coefficient of the leading term.
What is Not a Polynomial?
- Terms with variables raised to negative, fractional, or variable exponents are not polynomials (e.g., x⁻⁷, a^(1/2), a^a).
- Expressions with variables in denominators or non-integer exponents are not polynomials.
Polynomials with Multiple Variables
- Polynomials can include more than one variable, as long as each variable in each term is raised to a nonnegative integer power (e.g., 5x⁶ + 9y³ + a).
Key Terms & Definitions
- Term — a single part of a polynomial separated by plus or minus signs.
- Coefficient — the number multiplying a variable in a term.
- Monomial — a polynomial with one term.
- Binomial — a polynomial with two terms.
- Trinomial — a polynomial with three terms.
- Degree (of a term) — the exponent value of the variable in that term.
- Degree (of a polynomial) — the highest degree among its terms.
- Standard form — a way of writing a polynomial with terms in descending order of degree.
- Leading term — the first term in the standard form of a polynomial.
- Leading coefficient — the coefficient of the leading term.
Action Items / Next Steps
- Practice identifying the degree, leading term, and standard form of given polynomials.
- Complete the following: Multiply monomials by polynomials exercises.