A Pisot polynomial has exactly one real root greater than 1 and all other roots inside the unit circle. The Pisot polynomial with the smallest possible real root is . This real value is called the plastic constant and is approximately 1.32472.

A Salem polynomial has exactly two real roots, and , with the other roots on the unit circle. The Salem polynomial with the smallest known real root is . This is called Lehmer's polynomial, with the real root . Whether this is the smallest possible such value is an unsolved problem.

A cyclotomic polynomial has all its roots on the unit circle. The first cyclotomic polynomial to have a coefficient other than or is the , which has two coefficients of . The is the first that has a coefficient different from , , ; it has three coefficients of .

This Demonstration features polynomials of each type, for polynomials with a reasonable number of terms and the real root less than 2. The roots are shown along with the unit circle and a vertical line at 2. The polynomial is shown under the graphic along with a plot of the coefficients.