BackIntroduction to Polynomials: Structure, Classification, and Operations
Study Guide - Smart Notes
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Polynomials: Definitions and Structure
What is a Polynomial?
A polynomial is an algebraic expression where variables have only positive integer exponents and a finite number of terms. Polynomials do not include variables with negative, fractional, or variable exponents, nor do they include variables in denominators.
Monomial: A polynomial with one term.
Binomial: A polynomial with two terms.
Trinomial: A polynomial with three terms.
Polynomial: An expression with one or more terms.
Example: is a polynomial with three terms.
Identifying Polynomials
To determine if an expression is a polynomial, check for the following:
All exponents on variables are whole numbers (0, 1, 2, ...).
No variables appear in denominators or under radicals.
No negative or fractional exponents.
Examples:
is a polynomial (all exponents are whole numbers).
is not a polynomial (variable in denominator).
is not a polynomial (fractional exponent).
is not a polynomial (negative exponent).
Writing Polynomials in Standard Form
Standard Form of a Polynomial
A polynomial is in standard form when its terms are written in descending order of exponents, and like terms are combined.
Degree: The highest exponent of the variable in the polynomial.
Leading Coefficient: The coefficient of the term with the highest degree.
Constant Term: The term without a variable.
Example:
Given , the degree is 2, the leading coefficient is 3, and the constant term is 4.
Adding and Subtracting Polynomials
Combining Like Terms
To add or subtract polynomials, combine like terms (terms with the same variable and exponent).
Adding Polynomials: Add the coefficients of like terms.
Subtracting Polynomials: Distribute the negative sign, then combine like terms.
Examples
Add:
Subtract:
Practice Problems
Summary Table: Key Terms in Polynomials
Term | Definition | Example |
|---|---|---|
Monomial | One term | |
Binomial | Two terms | |
Trinomial | Three terms | |
Degree | Highest exponent | Degree of is 3 |
Leading Coefficient | Coefficient of highest degree term | Leading coefficient of is 6 |
Constant Term | Term with no variable | Constant term of is 5 |
