Suppose we are speaking about polynomials with real coefficients and real variable. Such a polynomial has the form `p ( x ) = a x^4 + b x^3 + c x^2 + d x + ` e where a , b , c , d and e are fixed real numbers and x is the real variable.
A polynomial `p ( x ) ` is divisible by a monomial `x - a ` if and only if `a ` is a zero of `p ( x ) . ` If we can find a root of the given polynomial, we therefore can factor it partially using polynomial division, then find a root of the remaining polynomial and so on.
But not all polynomials of even degree have a root under reals. For example, `x^2 + 1` is always positive. Because of this, a 4th degree polynomial may be a product of two 2nd degree polynomials without roots. Grouping may help to find such factoring, for example
`x^4 + 3 x^2 + 2 = x^4 + x^2 + 2x^2 + 2 = x^2 ( x^2 + 1 ) + 2 ( x^2 + 1 ) = ( x^2 + 1 ) ( x^2 + 2 ) .`
A polynomial with complex x always has 4 roots counted with multiplicity.