Other examples are solutions of linear differential equations with polynomial coefficients. The solutions of linear differential equations with polynomial coefficients are called holonomic functions. We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Sequences which are the solutions of linear difference equations with polynomial coefficients are called P-recursive. The remainder using polynomial arithmetic modulo 2 is formula 14. Calculating derivatives and integrals of polynomial functions is particularly simple. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial. These include the quintic equations defined by a polynomial that is reducible, such as . Similar matrices have the same characteristic polynomial. Two similar matrices have the same characteristic polynomial. The roots of the characteristic polynomial formula 120 are the eigenvalues formula 121. The eigenvalues are the roots of the characteristic polynomial. Note that formula 178 is a trigonometric polynomial of degree formula 23. Entries are 8-bit bytes treated as coefficients of polynomial of order formula 8. Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients.