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Irreducible (Prime) Polynomials

A polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial .

Example 1:

x 2 + x + 1

is an irreducible polynomial. There is no way to find two integers b and c such that their product is 1 and their sum is also 1 , so we cannot factor into linear terms ( x + b ) ( x + c ) .

Example 2:

The polynomial

x 2 2

is irreducible over the integers. However, we could factor it as

( x 2 ) ( x + 2 )

if we are allowed to use irrational numbers. So the irreducibility of a polynomial depends on the number system you're working in.

(When you study complex numbers , you'll find that the only irreducible polynomials over C are the degree 1 polynomials!)