Introduction to Polynomials

The parts of a polynomial

The coefficient is the number in front of a term: in $4x^{3}$, the coefficient is $4$. A term with no visible coefficient, like $x^{2}$, has a coefficient of $1$.

The degree of a polynomial is its highest exponent. In $5x^{4} + 3x^{2} - x + 8$ the exponents are $4$, $2$, $1$, and $0$, so the degree is $4$.

The leading coefficient is the coefficient of the highest-degree term. Standard form lists terms from highest degree down to the constant — once a polynomial is in standard form, the leading coefficient is simply the first number you see.

Naming polynomials by their number of terms

Polynomials get names based on how many terms they have. One term is a monomial, like $9x^{5}$. Two terms is a binomial, like $x^{2} - 16$. Three terms is a trinomial, like $x^{2} + 5x + 6$. Anything with more terms is usually just called a polynomial.

To count terms, count the pieces separated by $+$ or $-$ signs. The sign belongs to the term that follows it, so $4x^{3} + 2x - 7$ has the terms $4x^{3}$, $2x$, and $-7$.

What is not a polynomial

Every exponent in a polynomial must be a whole number: $0, 1, 2, 3, \ldots$ That rules out negative exponents like $x^{-2}$, variables inside roots like $\sqrt{x}$ (which means $\displaystyle x^{\frac{1}{2}}$), and variables in a denominator like $\dfrac{3}{x}$. If you spot any of those, the expression is not a polynomial.

Worked examples

Try one yourself

Common questions