Question

fractorize using synthetic division.
2x³-3x²-3x+2​

Answer 1

Step-by-step explanation:

Firstly, you can easily verify that f(2)<0<f(3), and hence there is a solution x between 2 and 3. In order to find it change the variable x to z=x-1/2, or x=z+1/2. The polynomial will turn into g(z)=2*z^3-(9/2)*z-4, After dividing by 2 you would like to solve the equation z^3-(9/4)*z-2=0. That cubic is of the form z^3–3pz-2q=0 with p=3/4 and q=1. Now you may apply the well known Cardano’s formula for the solution of that cubic to get:

z=(q+sqrt(q^2-p^3))^(1/3)+((q-sqrt(q^2-p^3))^(1/3)

to which you add 1/2 to get the answer

x=z+1/2

which is about

x=2.328589413348392155766239029667