Polynomials in Algebra

A polynomial is an algebraic expression containing powers of variable(s) and constants, connected only by addition, subtraction, and multiplication. The exponent of the variable is always a non-negative integer.

This polynomial has four terms: 2x³, -5x², 3x, and -7. The highest power (3) is the degree of the polynomial.

Parts of a Polynomial

• Term: a part of the polynomial, e.g., 2x³.
• Coefficient: the number before the variable, e.g., -5 in -5x².
• Degree: the highest exponent that appears in the polynomial.
• Constant term: a term that does not contain a variable, e.g., -7.

Naming Polynomials by Degree

• Degree 0: constant polynomial (e.g., 5).
• Degree 1: linear polynomial (e.g., 3x + 1).
• Degree 2: quadratic polynomial (e.g., x² - 4).
• Degree 3: cubic polynomial (e.g., x³ + 2x).
• Degree 4: quartic polynomial, and so on.

Operations with Polynomials

We can perform many operations with polynomials: addition, subtraction, multiplication, division (with remainder), and evaluation at a given point.

Notable Polynomials

• Binomial: polynomial with two terms, e.g., x + 5.
• Trinomial: polynomial with three terms, e.g., x² + 3x + 2.
• Monomial: polynomial with a single term, e.g., 7x³.

Polynomials in Everyday Life

• Physical formulas often appear in polynomial form, e.g., distance-time relationships.
• Economic models (e.g., cost functions) use polynomials to describe trends.
• In computer science, polynomials are used to analyze algorithms (e.g., running time as a polynomial function of input size).

Practice Exercise