For example, over the integers modulo , the derivative of the polynomial is the polynomial .

The checksum is generated using a polynomial value of 0x04C11DB7.

Thus if our polynomial is formula_10then the possible rational roots are all the integer divisors of a 0 ( 2) formula_11(This example is simple because the polynomial is monic (i.e.

In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial.

A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Another technique used to control ones density is the use of a scrambler polynomial on the raw data which will tend to turn the raw data stream into a stream that looks pseudo random, but where the raw stream can be recovered exactly by reversing the effect of the polynomial.

For any non zero polynomial formula_6 over the complex numbers in one variable, the solution set is made up of finitely many points.

For example, formula_1 is a homogeneous polynomial of degree 5, in two variables the sum of the exponents in each term is always 5.

For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.

So if formula_18 is our polynomial and formula_19 are the roots we have found, then consider the product formula_20By the fundamental theorem of algebra, R ( x ) should be equal to P ( x ), if all the roots of P ( x ) are rational.

The sign and relation of these roots, formula_28, to each other may be used to determine the stability of the dynamical system formula_29For a 2 dimensional system, the characteristic polynomial is of the form formula_30 where formula_31 is the trace and formula_32 is the determinant of A.