# How To Factor 4th Degree Polynomials?

To factor a polynomial of 4th degree, find its roots. If there are no roots, use grouping to factor it into the product of two second degree polynomials without roots. 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.

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