Polynomials: Classification, Standard Form, and Operations

Definition and Classification of Polynomials

A polynomial is an algebraic expression where variables have only whole number exponents and no negative exponents or fractions. Polynomials are classified based on the number of terms they contain.

• Monomial: An expression with 1 term. Example:

• Binomial: An expression with 2 terms. Example:

• Trinomial: An expression with 3 terms. Example:

Example: is a trinomial because it has three terms.

Identifying Polynomials and Their Types

To determine if an expression is a polynomial, check that all exponents are whole numbers and that there are no variables in denominators or under radicals. Then, count the number of terms to classify as monomial, binomial, trinomial, or none.

• Whole number exponents? Yes: Proceed. No: Not a polynomial.

• Number of terms: 1 (monomial), 2 (binomial), 3 (trinomial), more (polynomial), none (not a polynomial).

Example:

• is a monomial (one term, all exponents are whole numbers).

• is a binomial (two terms).

• is a trinomial (three terms).

Writing Polynomials in Standard Form

Polynomials are written in standard form by arranging terms in descending order of exponents. Terms with the same variable and exponent are called like terms and should be combined.

• Degree: The highest exponent of the variable in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

Example: For :

• Degree: 2 (from )

• Leading Coefficient: 3 (coefficient of )

• Constant: 4 (term without a variable)

Examples: Writing in Standard Form

• Arrange terms in descending order of exponents.

• Combine like terms if necessary.

• Identify the degree and leading coefficient.

Example A: Descending order: $\frac{1}{2}x^2 + x$ Degree: 2 Leading Coefficient:

Example B: Combine like terms: Standard form: Degree: 2 Leading Coefficient: -2

Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms (terms with the same variable and exponent). When subtracting, distribute the negative sign to each term in the second polynomial before combining.

• Adding: Combine like terms: , , Result:

• Subtracting: Distribute the negative: Combine like terms: , , Result:

Caution: Always distribute the negative sign when subtracting polynomials.

Practice Problems

• Example 1: Combine like terms: , , , $4

• Example 2: Distribute the negative: Combine like terms: , , , Result:

Summary Table: Types of Polynomials