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Question

What is irreducible factors?


Solution

It is kind of quadratic polynomial which cannot be split into two linear polynomials with real coefficients. So the one that cannot be written as: 
F(x)=(ax+b)(cx+d)F(x)=(ax+b)(cx+d) where a,b,c,d

We can derive another definition.

Let's rewrite it as:
F(x) = a(x+b/a)c(x+d/c) = 
= ac(x-(-b/a))(x-(-d/c))

This polynomial is equal to 0 when x equals to -b/a or -d/c . So these numbers are the roots of this polynomial. 
That's why we can formulate our definition that the irreducible quadratic polynomial is the one that doesn't have real roots.

For example we can carry out reduction on:
x^2 - 5x + 6  =
= (x-2)(x-3)

However we cannot do this with this one: 
x^2 + 1 

It doesn't have real roots.

Mathematics

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