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A polynomial in one variable x is a function f(x) of the form:

f(x) = anxn + an − 1xn − 1 + ... + a2x2 + a1x + a0

In elementary mathematics, the coefficients ai are typically chosen to be real or complex numbers. However, it makes sense to define a polynomial with coefficients in any ring.

The largest power of x that appears in the polynomial is called the degree of the polynomial.

  • Example: f(x) = 4x3 − 3x + 1 is a degree 3 polynomial with integer coefficients.

A polynomial in two variables x,y is, similarly, a finite sum

\sum a_{ij}x^i y^j
, where the coefficients aij are elements of some ring. Polynomials in 3 or more variables are defined similarly.

When we substitute a value b into the polynomial f(x) then we get an actual number as output. This number is the value anbn + an − 1bn − 1 + ... + a2b2 + a1b + a0. This process is called "evaluating the polynomial at b".

Some algorithms are said to perform in polynomial time. There are algorithms which can factor polynomials in an amount of time that is a polynomial nk.

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