Notice that:
[tex]F(3)=3^3-3^2-4\cdot3-6=27-9-12-6=27-27=0.[/tex]
Therefore 3 is a root of the given polynomial.
Now, we can use this root to factor the polynomial:
[tex]F(x)=(x-3)\frac{x^3-x^2-4x-6}{x-3}.[/tex]
Using the synthetic division algorithm we get that:
[tex]\frac{x^3-x^2-4x-6}{x-3}=x^2+2x+2.[/tex]
The roots of the above polynomial are:
[tex]\begin{gathered} x=-1+i, \\ x=-1-i\text{.} \end{gathered}[/tex]
Therefore:
[tex]F(x)=\mleft(x-3\mright)(x+1+i)(x+1-i)\text{.}[/tex]
Answer:
[tex]F(x)=(x-3)(x+1+i)(x+1-i)\text{.}[/tex]
