Factor the polynomial function over the complex numbers? ? f(x)= x^4 -x^3-2x-4

The factor form of the function F(x) is , where   gives two complex roots.

Explanation:

The given function is,

According to the rational root theorem 1 and -1 are the possible roots of each polynomial.

Put x=-1

Since the value of F(x) is 0 at x=-1 therefore the -1 is a root of given polynomial and (x+1) is a factor of F(x).

Use synthetic division method to divide the polynomial by (x+1).

The last row of the synthetic division shows the coefficient of remaining polynomial.

At x=2 the value of F(x) is 0, therefore (x-2) is a factor of F(x).

Use synthetic division method to divide the polynomial by (x-2).

Therefore the roots of F(x) are,

Where factors with real root and are factors with complex roots.

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