Polynomials: Definitions and Degree

What makes an expression a polynomial

A polynomial is an algebraic expression where every variable is raised only to a non-negative whole-number power. That single rule is what disqualifies some expressions that otherwise look similar:

• $4x^2 - 15x + 9$ is a polynomial (powers 2, 1, 0, all non-negative whole numbers).
• $5x + 2y^{-1} - 120$ is not a polynomial: $y$ has a negative power.
• $y^{6/7} + 2z - \tfrac{1}{2}$ is not a polynomial: $6/7$ isn't a whole number.

Monomial, binomial, trinomial

The parts of a polynomial separated by $+$ or $-$ are its terms. The number of terms names the expression:

• Monomial: one term, e.g. $5z$ or $4x^2$.
• Binomial: two terms, e.g. $3p - 2q$.
• Trinomial: three terms, e.g. $4x^2 + 6x + 7$.

Finding the degree

In one variable, the degree is just the highest power present. With multiple variables, add the exponents within each term, then take the highest total across all terms.

Worked example: find the degree of $5x^2y^3 + 2x^2y^2 - 8xy$.

Add exponents within each term:

The highest total is 5, so the polynomial has degree 5.