Question

Which of the following options correctly represents the complete factored form of the polynomial F(x)= x - x2 - 4x-6?

Answer 1

Notice that:

`F(3)=3^3-3^2-4\cdot3-6=27-9-12-6=27-27=0.`

Therefore 3 is a root of the given polynomial.

Now, we can use this root to factor the polynomial:

`F(x)=(x-3)(x^3-x^2-4x-6)/(x-3).`

Using the synthetic division algorithm we get that:

`(x^3-x^2-4x-6)/(x-3)=x^2+2x+2.`

The roots of the above polynomial are:

`\begin{gathered} x=-1+i, \\ x=-1-i\text{.} \end{gathered}`

Therefore:

`F(x)=\mleft(x-3\mright)(x+1+i)(x+1-i)\text{.}`

Answer:

`F(x)=(x-3)(x+1+i)(x+1-i)\text{.}`